Physics Higher Unit 1. Note: The Radiation and Matter component of Higher Physics will appear on the following Science CD-ROM.

Transcription

1 Physics Higher Unit Note: The Radiation and Matter component of Higher Physics will appear on the following Science CD-ROM.

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3 Spring 1999 HIGHER STILL Physics Higher Support Materials

4 This publication may be reproduced in whole or in part for educational purposes provided that no profit is derived from the reproduction and that, if reproduced in part, the source is acknowledged. First published 1999 Higher Still Development Unit PO Box Ladywell House Ladywell Road Edinburgh EH12 7YH

5 HIGHER PHYSICS - STAFF NOTES Checklist Higher Physics course and units The Higher Physics course is divided into the following three units. Mechanics and Properties of Matter (40 hours) Electricity and Electronics (40 hours) Radiation and Matter (40 hours) The support materials For each of the three units the student material contains the following sections. Checklist Summary Notes Activities Problems (with numerical answers) In addition there is a separate section dealing with uncertainties, units and prefixes. It should be noted that the Content Statements associated with uncertainties are part of each of the three units, see the Arrangements for Physics. Although the Higher Physics units could be taught in any order, current practice indicates that the Mechanics and Properties of Matter is usually taught first. Hence the section dealing with uncertainties is placed at the start of this unit. The examples in the section on uncertainties are taken from topics in this unit. The problem sections of the other two units contain a few additional questions on uncertainties. The student materials are to provide assistance to the teacher or lecturer delivering a unit or the course. They are not a self standing open learning package. They require to be supplemented by learning and teaching strategies. This is to ensure that all the unit or course content is covered and that the students are given the support they need to acquire the necessary knowledge, understanding and skills demanded by the unit or the course. Checklists These are lists of the content statements taken directly from the Arrangements for Physics documentation. Summary Notes These notes are a brief summary of all the essential content and include a few basic worked examples. They are intended to aid students in their revision for unit and course assessment. Explanation of the concepts and discussion of applications are for the teacher or lecturer to include as appropriate. Physics: Higher Staff notes 5

6 Checklist Activities The activity pages provide suggestions for experimental work. A variety of practical activities have been included for each unit. The instruction sheet can be adapted to suit the equipment available in the centre. Some activities are more suitable for teacher demonstration and these have been included for use where appropriate. For each unit there is a range of activities. Although it is desirable that students are able to undertake practical investigations, it must be mentioned that this does not imply that all activities must be undertaken. The teacher or lecturer should decide what to select depending on the needs of their students and the learning and teaching approaches adopted. For Outcome 3 of each unit one report of a practical activity is required. Some activities suitable for the achievement of Outcome 3 have been highlighted and should be seen as an opportunity to develop good practice. Problems A variety of problems have been collated to give the student opportunity for practice and to aid the understanding of the unit or course content. Use of the materials The checklists may be issued at the end or at the beginning of a unit depending on the teacher s discretion. Hence for each unit the checklists are given with separate page numbers. The rest of the material for each unit is numbered consecutively through the summary notes, activities and problems. The uncertainties section is numbered separately. Some staff may wish to cover this material at the start of the course, others may prefer to introduce the concepts more gradually during early experimental work. (For unit assessment, uncertainties are covered in Outcome 3. The course assessment will contain questions which will sample uncertainties within the context of any of the units.) When photocopying a colour code for each section could be an advantage. For some units it could be preferable to split the unit into subsections. For example, the Mechanics and Properties of Matter unit might have a first subsection booklet for kinematics which would entail selecting appropriate summary notes, activities, problems and numerical answers. Learning and Teaching A variety of teaching methods can be used. Direct teaching whether it be to a whole class or small groups is an essential part of the learning process. A good introduction to a topic; for example, a demonstration, activity or video, is always of benefit to capture the minds of the students and generate interest in the topic. Applications should be mentioned and included wherever possible. Physics: Higher Staff notes 6

7 Checklist Further materials A course booklet will be issued which will contain additional problems including: examination type questions of a standard suitable for estimates of course performance and evidence for appeals revision home exercises for ongoing monitoring of progress. Outcome 3 The Handbook: Assessing Outcome 3 Higher Physics contains specific advice for this outcome together with exemplar instruction sheets and sample student reports. Specimen Course Assessment A specimen course question paper together with marking scheme has been issued by SQA. This pack also contains the updated Details of the instrument for external assessment from the Arrangements for Physics. Physics: Higher Staff notes 7

8 MECHANICS AND PROPERTIES OF MATTER Checklist The knowledge and understanding content for this unit is given below. Vectors 1. Distinguish between distance and displacement. 2. Distinguish between speed and velocity. 3. Define and classify vector and scalar quantities. 4. Use scale diagrams, or otherwise, to find the magnitude and direction of the resultant of a number of displacements or velocities. 5. State what is meant by the resultant of a number of forces. 6. Carry out calculations to find the rectangular components of a vector. 7. Use scale diagrams, or otherwise, to find the magnitude and direction of the resultant of a number of forces. Equations of motion 1. State that acceleration is the change in velocity per unit time. 2. Describe the principles of a method for measuring acceleration. 3. Draw an acceleration-time graph using information obtained from a velocity-time graph for motion with a constant acceleration. 4. Use the terms constant velocity and constant acceleration to describe motion represented in graphical or tabular form. 5. Show how the following relationships can be derived from basic definitions in kinematics: v = u + at s = ut at v = u + 2as 2 6. Carry out calculations using the above kinematic relationships. Newton s Second Law, energy and power 1. Define the newton. 2. Carry out calculations using the relationship F = ma in situations where resolution of forces is not required. 3. Use free body diagrams to analyse the forces on an object. 4. Carry out calculations involving work done, potential energy, kinetic energy and power. Physics: Mechanics and Properties of Matter (H) 1

9 Checklist Momentum and impulse 1. State that momentum is the product of mass and velocity. 2. State that the law of conservation of linear momentum can be applied to the interaction of two objects moving in one dimension, in the absence of net external forces. 3. State that an elastic collision is one in which both momentum and kinetic energy are conserved. 4. State that an inelastic collision is one in which only momentum is conserved. 5. Carry out calculations concerned with collisions in which the objects move in only one dimension. 6. Carry out calculations concerned with explosions in one dimension. 7. Apply the law of conservation of momentum to the interaction of two objects moving in one direction to show that: a) the changes in momentum of each object are equal in size and opposite in direction b) the forces acting on each object are equal in size and opposite indirection. 8. State that impulse = force time. 9. State that impulse = change in momentum. 10. Carry out calculations using the relationship, impulse = change of momentum. Density and Pressure 1. State that density is mass per unit volume. 2. Carry out calculations involving density, mass and volume. 3. Describe the principles of a method for measuring the density of air. 4. State and explain the relative magnitudes of the densities of solids, liquids and gases. 5. State that pressure is the force per unit area, when the force acts normal to the surface. 6. State that one pascal is one newton per square metre. 7. Carry out calculations involving pressure, force and area. 8. State that the pressure at a point in a fluid at rest is given by h g. 9. Carry out calculations involving pressure, density and depth. 10. Explain buoyancy force (upthrust) in terms of the pressure difference between the top and bottom of an object. Gas laws 1. Describe how the kinetic model accounts for the pressure of a gas. 2. State that the pressure of a fixed mass of gas at constant temperature is inversely proportional to its volume. 3. State that the pressure of a fixed mass of gas at constant volume is directly proportional to its temperature measured in kelvin K. 4. State that the volume of a fixed mass of gas at constant pressure is directly proportional to its temperature measured in kelvin K. 5. Carry out calculations to convert temperatures in o C to K and vice versa. 6. Carry out calculations involving pressure, volume and temperature of a fixed mass of gas using the general gas equation. 7. Explain what is meant by absolute zero of temperature. 8. Explain the pressure-volume, pressure-temperature and volume-temperature laws qualitatively in terms of the kinetic model. Physics: Mechanics and Properties of Matter (H) 2

10 Checklist Uncertainties 1. State that measurement of any physical quantity is liable to uncertainty. 2. Distinguish between random uncertainties and recognised systematic effects. 3. State that the scale reading uncertainty is a measure of how well an instrument scale can be read. 4. Explain why repeated measurements of a physical quantity are desirable. 5. Calculate the mean value of a number of measurements of the same physical quantity. 6. State that this mean is the best estimate of a true value of the quantity being measured. 7. State that where a systematic effect is present the mean value of the measurements will be offset from a true value of the physical quantity being measured. 8. Calculate the approximate random uncertainty in the mean value of a set of measurements using the relationship: approximate random uncertainty in the mean = maximum value - minimum value number of measurements taken 9. Estimate the scale-reading uncertainty incurred when using an analogue display and a digital display. 10. Express uncertainties in absolute or percentage form. 11. Identify, in an experiment where more than one physical quantity has been measured, the quantity with the largest percentage uncertainty. 12. State that this percentage uncertainty is often a good estimate of the percentage uncertainty in the final numerical result of the experiment. 13. Express the numerical result of an experiment in the form: final value ± uncertainty. Units, prefixes and scientific notation 1. Use SI units of all physical quantities appearing in the Content Statements. 2. Give answers to calculations to an appropriate number of significant figures. 3. Check answers to calculations. 4. Use prefixes (p, n, µ, m, k, M, G). 5. Use scientific notation. Physics: Mechanics and Properties of Matter (H) 3

11 UNCERTAINTIES Uncertainties It is important to realise a degree of uncertainty is associated with any measured physical quantity. Systematic Effects These can occur when the measurements are affected all in the same way e.g. a metre stick might have shrunk, thus giving consistently incorrect readings. At this level, the systematic effect tends to be small enough to be ignored. Where accuracy is of the utmost importance, the apparatus would be calibrated against a known standard. Note that a systematic effect might also be present if the experimenter is making the same mistake each time in taking a reading. Random Uncertainty Random fluctuations can affect measurements from reading to reading, e.g. consecutive timings of the period of a pendulum can differ. The best estimate of the true value is given by repeating the readings and then calculating the mean value. The random uncertainty is then calculated using the formula below. Random uncertainty = maximum reading - minimum reading number of readings Scale Reading Uncertainty This value indicates how well an instrument scale can be read. An estimate of reading uncertainty for an analogue scale is generally taken as: ± half the least division of the scale. Note: for widely spaced scales, this can be a little pessimistic and a reasonable estimate should be made. For a digital scale it is taken as ± 1 in the least significant digit displayed. Examples 6 7 cm 0 10 m 8.94 s ( ) cm Length lies between (6.55 and 6.65) cm ( ) m Length lies between (8.5 and 9.5) m ( ) s Time lies between (8.93 and 8.95) s. Physics: Uncertainties and Prefixes (H) 1

12 Example The times for 10 swings of a pendulum are: 1.1, 1.4, 1.2, 1.3 and 1.1 s Uncertainties Mean value = 1.2 s Random uncertainty = maximum - minimum number of readings = = 0.06 s Time for 10 swings = (1.2 ± 0.1) s = 1.2 s ± 5 % Note: when the uncertainty is expressed in units then it is known as the absolute uncertainty. In this case this is s, or ± 0.1 s. Comparison of Uncertainties When comparing uncertainties, it is important to take the percentage in each. Suppose in an experiment the following uncertainties were found. Systematic = 0.1 % Scale Reading = 2 % Random = 0.5 % The overall uncertainty should be taken as the highest percentage uncertainty. In this case, this would be the reading uncertainty at 2 %. Note: since accuracy is now being quantified, it is essential when using a calculator that all the figures are not taken down, since every number stated indicates the degree of accuracy. As a general rule, your answer should contain the same number of significant figures as the least accurate reading. Examples 1. Refer to the example at the top of the page. The mean value is 1.22 s and the random uncertainty 0.06 s. However, all the readings are to two significant figures hence the final answer must be written as (1.2 ± 0.1)s as shown. 2. Calculate the average speed and absolute uncertainty from the following readings. s = (1.54 ± 0.02) m % uncertainty in s = t = (1.69 ± 0.01) s % uncertainty in t = = 1.3 % = 0.6 % Highest uncertainty taken = 1.3 % v = s = 1.54 = ms % t % of m s -1 = m s -1 (converts % to absolute uncertainty) v = (0.91 ± 0.01) m s -1 Physics: Uncertainties and Prefixes (H) 2

13 ACTIVITY Uncertainties Title: Uncertainties Aim: to find the average speed of a trolley moving down a slope, estimating the uncertainty in the final value. Apparatus: 1 ramp, 1 metre stick, 1 trolley, 1 stop clock. Instructions Set up a slope and mark two points 85 cm apart. Note the scale reading uncertainty. Calculate the percentage uncertainty in the distance. Ensuring the trolley starts from the same point each time, measure how long it takes the trolley to pass between the two points. Repeat 5 times, calculate the mean time and estimate the random uncertainty. Note the scale reading uncertainty in the time. Calculate the percentage uncertainty in the time. Calculate the average speed and associated uncertainty. Express your result in the form: (speed ± absolute uncertainty) m s -1 Physics: Uncertainties and Prefixes (H) 3

14 Problems Uncertainties 1. Calculate the percentage uncertainties for the following absolute readings: a) (4.65 ± 0.05) V b) (892 ± 5) cm c) (1.8 ± 0.4) A d) (2.87 ± 0.02) s e) (13.8 ± 0.5) Hz f) (5.2 ± 0.1) m. 2. State the three types of uncertainty, explaining the difference between them. 3. Manufacturers of resistors state the uncertainty in their products by using colour codes. Gold - 5 % accuracy. Silver - 10 % accuracy. Calculate the possible ranges for the following resistors for each colour. a) 1 k b) 10 k c) For each of the following scales, write down the reading and estimate the uncertainty. 5. Calculate the mean time and random uncertainty for the following readings: 0.8 s, 0.6 s, 0.5 s, 0.6 s and 0.4 s. 6. A student uses light gates and suitably interfaced computer to measure the acceleration of a trolley as it moves down a slope. The following results were obtained. a / m s , 5.24, 5.21, 5.19, 5.12, 5.20, 5.17, Calculate the mean acceleration and the corresponding random uncertainty. 7. AB is measured using a metre stick. A trolley is timed between AB. The following results were obtained. AB = ( ) cm t/s 1.21, 1.21, 1.26, 1.27, 1.24 and Express the average speed in the form (value ± absolute uncertainty). Physics: Uncertainties and Prefixes (H) 4

15 Solutions Uncertainties 1. (a) uncertainty = = 1.1 % (b) 0.6 % (c) 22 % (d) 0.7 % (e) 3.6 % (f) 1.9 % 2. Systematic effect: affects all readings in the same way. Can be due to apparatus limitations or fault in experimental approach. Reading uncertainty: accuracy limited by quality of scale ± half the least division (analogue) or good estimate. ± 1 in the least significant digit (digital). Random uncertainty: random fluctuations between readings. Effect is minimised by repeating readings. Uncertainty = maximum reading - minimum reading number of readings 3. (a) Gold 5 % of 1k = 0.05 k Silver 10 % of 1k = 0.1k (b) Silver ( ) (c) Silver ( ) Range is ( ) k Range is ( ) k Gold ( ) Gold ( ) = ( ) = ( ) 4. (a) ( 3.2 ± 0.1) mm (b) (2.30 ± 0.05) cm (c) (0.250 ± 0.005) 0 C (d) (2.4 ± 0.2) g (e) (7.84 ± 0.01) s (f) (1.005 ± 0.001) s (g) (195 ± 1) ma 5. ( ) Mean time = 5 ( ) Random uncertainty = = 0.08 s 5 t = (0.6 ± 0.1) s = 0.6 s 6. Mean a = 5.19 m s Random error = = ms -2 8 a = (5.19 ± 0.02) m s AB = (60.0 ± 0.1) cm = 60 cm ± 0.17 % Random uncertainty in t = 6 Mean t = (1.25 ± 0.01) s = 1.25 s ± 0.8 % = s Greatest percentage uncertainty is 0.8 % in the time. v = s t = ± 0.8% = 48.0 cm s-1 ± 0.8% = (48.0 ± 0.4) m s -1 Physics: Uncertainties and Prefixes (H) 5

16 PREFIXES Prefixes The following are prefixes used to denote multiples and sub-multiples of any unit used to measure a physical quantity. Name Symbol Power of 10 tera giga T G mega M 10 6 kilo k 10 3 multiples centi c 10-2 milli m 10-3 micro µ 10-6 sub-multiples nano n 10-9 pico p Questions Use scientific notation to write the measurements in the units shown gigahertz = 12 GHz = Hz megohms = 4.7 M = kilometres = 46 km = m millivolts = 3.6 mv = V milliamps = 0.55 ma = A microamps = 25 µa = A nanometres = 630 nm = m picofarads = 2200 pf = F Rewrite the following quantities in the units shown m = km = M Hz = GHz = MHz V = mv = µv A = µa = ma m = nm F = pf = µf Physics: Uncertainties and Prefixes (H) 6

17 VECTORS Summary Notes Distance and Displacement. Distance is the total path length. It is fully described by magnitude (size) alone. Displacement is the direct length from a starting point to a finishing point. To describe displacement both magnitude and direction must be given. Example A woman walks 3 km due North (000) and then 4 km due East (090). Find her a) distance travelled b) displacement i.e. how far she is from where she started? Using a scale of 1cm: 1km draw an accurate scale diagram as shown above. a) Distance travelled = AB + BC = 7 km b) Measuring AC = 5 cm. Convert using the scale gives the magnitude of the displacement = 5 km Use a protractor to check angle BAC = 53 o that is 53 o east of north. Displacement = 5 km (053) Speed and Velocity These two quantities are fundamentally different. Average speed = distance time Average velocity = displacement time Velocity has an associated direction, being the same as that of the displacement. The unit for both these quantities is metres per second, m s -1. Vectors and Scalars A scalar quantity is completely defined by stating its magnitude. A vector quantity is completely defined by stating its magnitude and direction. Examples are given below. Physics: Mechanics and Properties of Matter (H) Student Material 1

18 Addition of Vectors When vectors are being added, their magnitude and direction must be taken into account. This can be done using a scale diagram and adding the vectors tip to tail, then joining the starting and finishing points. The final sum is known as the resultant, the single vector that has the same effect as the sum of the individuals. Example Find the resultant force acting at point O. Step 1: Choose a suitable scale, e.g. 1 cm to 1 N. Step 2: Arrange arrows tip to tail. Step 3: Draw in resultant vector, measuring its length and direction. 1 cm to 1 N. AC = 12.2 cm Force = 12.2 N Using a protractor, angle BAC measures 12 0 Bearing = = Resultant Force = 12.2 N at (102) Vectors at right angles If the vectors are at right angles then it may be easier to use Pythagoras to find the resultant and trigonometry to find an angle. Addition of more than two vectors Use a scale diagram and ensure that each vector is placed tip to tail to the previous vector. The resultant vector is the vector from the starting point to the finishing point in magnitude and direction. Resultant of a number of forces The resultant of a number of forces is that single force which has the same effect, in both magnitude and direction, as the sum of the individual forces. Physics: Mechanics and Properties of Matter (H) Student Material 2

19 Rectangular components of a vector Resolution of a vector into horizontal and vertical components. Any vector v can be split up into a horizontal component v h and vertical component v v. is equivalent to Example A shell is fired from a cannon as shown. Calculate its a) horizontal component of velocity b) vertical component of velocity. a) v h = v cos u = 50 cos 60 = 25 m s -1 b) v v = v sin u = 50 sin 60 = 43 m s -1 So is equivalent to Physics: Mechanics and Properties of Matter (H) Student Material 3

20 EQUATIONS OF MOTION Acceleration Acceleration is defined as the change in velocity per unit time. The unit is metre per second squared, m s -2. a = v - u t where v = final velocity u = initial velocity t = time taken Measuring acceleration Acceleration is measured by determining the initial velocity, final velocity and time taken. A double mask which interrupts a light gate can provide the data to a microcomputer and give a direct reading of acceleration. Acceleration-time and velocity-time graphs Constant velocity v / m s -1 Constant positive acceleration (velocity increasing) v / m s -1 Constant deceleration Constant negative acceleration (velocity decreasing) v / m s -1 t / s t / s t / s a / m s -2 a / m s -2 a / m s -2 t / s t / s t / s a = 0 acceleration is positive acceleration is negative Constant velocity and constant acceleration The velocity time graph below illustrates these terms. OA AB BC is constant acceleration, the acceleration is positive. is constant velocity, the acceleration is zero. is constant deceleration, the acceleration is negative. Physics: Mechanics and Properties of Matter (H) Student Material 4

21 Equations of motion v = u + at s = ut at2 where: u - initial velocity of object at time t = 0 v - final velocity of object at time t a - acceleration of object t - time to accelerate from u to v s - displacement in time t. v 2 = u 2 + 2as These equations of motion apply providing: the motion is in a straight line the acceleration is uniform. When using the equations of motion, note: the quantities u, v, s and a are all vector quantities a positive direction must be chosen and quantities in the reverse direction must be given a negative sign a deceleration will be negative, for movement in the positive direction. Derivation of equations of motion The velocity - time graph for an object accelerating uniformly from u to v in time t is shown below. a = v - u t Changing the subject of the formula gives: v = u + at [1] The displacement s in time t is equal to the area under the velocity time graph. Area = Area of triangle A + area of rectangle B s = 1 (v-u)t + ut 2 but from equation 1, v-u = at s = 1 2 (at)t + ut s = ut at [2] Physics: Mechanics and Properties of Matter (H) Student Material 5

22 Using v = u + at, v 2 = (u + at) 2 v 2 = u 2 + 2uat + a 2 t 2 v 2 = u 2 + 2a(ut at2 ) Since s = ut at2 v 2 = u 2 + 2as [3] Physics: Mechanics and Properties of Matter (H) Student Material 6

23 Projectile motion A projectile has a combination of vertical and horizontal motions. Various experiments show that these horizontal and vertical motions are totally independent of each other. Closer study gives the following information about each component. Horizontal: constant speed Vertical: constant acceleration downward (due to gravity). Example An object is released from an aircraft travelling horizontally at 1000 m s -1. The object takes 40 s to reach the ground. a) What is the horizontal distance travelled by the object? b) What was the height of the aircraft when the object was released? c) Calculate the vertical velocity of the object just before impact. d) Find the resultant velocity of the object just before hitting the ground. Before attempting the solution, you should divide your page into horizontal and vertical and enter appropriate information given or known. Horizontal Vertical v h = 1000 m s-1 t = 40 s t = 40 s uv = 0 a = 9.8 ms2 a) sh =? b) sv =? sv = ut + at2 sh = v t = = m = = m s-1 (downwards) c) vv =? = 7840 m = 392 m vv = u + at vv = 392 d) tanx = = 21o v2 = Physics: Mechanics and Properties of Matter (H) Student Material 7

24 NEWTON S SECOND LAW, ENERGY AND POWER Dynamics deals with the forces causing motion and the properties of the resulting moving system. Newton s 1st Law of Motion Newton s 1st law of Motion states that an object will remain at rest or travel with a constant speed in a straight line (constant velocity) unless acted on by an unbalanced force. Newton s 2nd Law Newton s 2nd law of motion states that the acceleration of an object: varies directly as the unbalanced force applied if the mass is constant varies inversely as the mass if the unbalanced force is constant. These can be combined to give F a m a kf m where k is a constant kf = ma The unit of force, the newton is defined as the resultant force which will cause a mass of 1kg to have an acceleration of 1 m s -2. Substituting in the above equation. k 1 = 1 1 k = 1 Provided F is measured in newtons, the equation below applies. F = ma m s -2 N kg Free Body Diagrams Some examples will have more than one force acting on an object. It is advisable to draw a diagram of the situation showing the direction of all forces present acting through one point. These are known as free body diagrams. Examples 1. On take off, the thrust on a rocket of mass 8000 kg is 200,000 N. Find the acceleration of the rocket. Thrust = 200,000 N Resultant force = ,400 = 121,600 N Weight = mg = = 78,400 N Physics: Mechanics and Properties of Matter (H) Student Material 8

25 a = F m = = 15.2 m s A woman is standing on a set of bathroom scales in a stationary lift (a normal everyday occurrence!). The reading on the scales is 500 N. When she presses the ground floor button, the lift accelerates downwards and the reading on the scales at this moment is 450 N. Find the acceleration of the lift. Weight = 500 N W = 500 N Force upwards = 500 N F = 450 N (reading on scales) Lift is stationary, forces balance W = F = 500 N Lift accelerates downwards, unbalanced force acts. Resultant Force = Weight - Force from floor = W - F = = 50 N a = Resultant Force m = = 1 m s Tension A ski tow pulls 2 skiers who are connected by a thin nylon rope along a frictionless surface. The tow uses a force of 70 N and the skiers have masses of 60 kg and 80 kg. Find a) the acceleration of the system b) the tension in the rope. 60 kg 80kg a) Total mass, m = 140 kg a = F m = 70 = 0.5 m s b) Consider the 60 kg skier alone. Tension, T = ma = = 30 N Physics: Mechanics and Properties of Matter (H) Student Material 9

26 Resolution of a Force In the previous section, a vector was split into horizontal and vertical components. This can obviously apply to a force. is equivalent to F v = Fsin F h = Fcos Example A man pulls a garden roller of mass 100 kg with a force of 200 N acting at 30 0 to the horizontal. If there is a frictional force of 100 N between the roller and the ground, what is the acceleration of the roller along the ground? F h = F cos = 200 cos 30 0 =173.2 N Resultant F h = Friction = = 73.2 N a = F m = = m s-2 Force Acting Down a Plane If an object is placed on a slope then its weight acts vertically downwards. A certain component of this force will act down the slope. The weight can be split into two components at right angles to each other. Component of weight down slope = mgsin Component perpendicular to slope = mgcos Example A wooden block of mass 2 kg is placed on a slope at 30 to the horizontal as shown. A frictional force of 4 N acts up the slope. The block slides down the slope for a distance of 3 m. Determine the speed of the block at the bottom of the slope. Component of weight acting down slope = mg sin30 = = 9.8 N Resultant force down slope = friction = = 5.8 N a = F/ v 2 = u 2 + 2as = 5.8 / 2 = Physics: Mechanics and Properties of Matter (H) Student Material 10

27 = 2.9 m s -2 = 17.4 v = 4.2 m s -1 Conservation of Energy The total energy of a closed system must be conserved, although the energy may change its form. The equations for calculating kinetic energy E k, gravitational potential energy E p and work done are given below. E p = mgh E k = 1 2 mv2 work done = force displacement Energy and work are measured in joules J. Example A trolley is released down a slope from a height of 0.3 m. If its speed at the bottom is found to be 2 m s -1, find a) the energy difference between the E p at top and E k at the bottom. b) the work done by friction c) the force of friction on the trolley a) E p at top = mgh = = 2.94 J E k at bottom = 1 2 mv2 = = 2 J Energy difference = 0.94 J b) Work done by friction = energy difference (due to heat, sound) = 0.94 J c) Work done = Force of friction d = 0.94 J d = 2 m F = 0.94 = 0.47 N 2 Force of friction = 0.47 N Power Power is the rate of transformation of energy from one form to another. P = energy time = work done time = F displacement t = F average velocity Power is measured in watts W. Physics: Mechanics and Properties of Matter (H) Student Material 11

28 MOMENTUM AND IMPULSE The momentum of an object is given by: Momentum = mass velocity of the object. Momentum = mv kg m s -1 kg m s -1 Note: momentum is a vector quantity. The direction of the momentum is the same as that of the velocity. Conservation of Momentum When two objects collide it can be shown that momentum is conserved provided there are no external forces applied to the system. For any collision: Total momentum of all objects before = total momentum of all objects after. Elastic and inelastic collisions An elastic collision is one in which both kinetic energy and momentum are conserved. An inelastic collision is one in which only momentum is conserved. Example a) A car of mass 1200 kg travelling at 10 m s -1 collides with a stationary car of mass 1000 kg. If the cars lock together find their combined speed. b) By comparing the kinetic energy before and after the collision, find out if the collision is elastic or inelastic. Draw a simple sketch of the cars before and after the collision. BEFORE AFTER a) Momentum = mv Momentum = mv = = ( ) v = kg m s -1 = 2200v kg m s -1 Total momentum before = Total momentum after = 2200v = v v = 5.5 m s -1 Physics: Mechanics and Properties of Matter (H) Student Material 12

29 b) Ek = 1 2 mv2 Ek = 1 2 mv2 = = = 60,000 J = 33,275 J Kinetic energy is not the same, so the collision is inelastic Vector nature of momentum Remember momentum is a vector quantity, so direction is important. Since the collisions dealt with will act along the same line, then the directions can be simplified by giving: momentum to the right a positive sign and momentum to the left a negative sign. Example Find the unknown velocity below. v = 4m s -1 v =? 8 kg 6 kg BEFORE Momentum = mv = (8 4) - (6 2) = 20 kg m s -1 (8 + 6 )kg AFTER Momentum = mv = (8 + 6)v = 14v Total momentum before = Total momentum after 20 = 14v v = = 1.43 m s-1 Trolleys will move to the right at 1.43 m s -1 since v is positive. Physics: Mechanics and Properties of Matter (H) Student Material 13

30 Explosions A single stationary object may explode into two parts. The total initial momentum will be zero. Hence the total final momentum will be zero. Notice that the kinetic energy increases in such a process. Example Two trolleys shown below are exploded apart. Find the unknown velocity. BEFORE AFTER Total momentum = mv Total momentum = mv = 0 = - (2x3) + 1v = v Total momentum before = Total momentum after 0 = -6 + v v = 6 m s -1 to the right (since v is positive). Impulse An object is accelerated by a force F for a time, t. The unbalanced force is given by: F = ma = m(v - u) t = mv - mu t Unbalanced force = change in momentum time Ft = mv - mu = rate of change of momentum The term Ft is called the impulse and is equal to the change in momentum. Note: the unit of impulse, Ns will be equivalent to kg m s -1. The concept of impulse is useful in situations where the force is not constant and acts for a very short period of time. One example of this is when a golf ball is hit by a club. During contact the unbalanced force between the club and the ball varies with time as shown below. Since F is not constant the impulse (Ft) is equal to the area under the graph. In any calculation involving impulse the unbalanced force calculated is always the average force and the maximum force experienced would be greater than the calculated average value. Physics: Mechanics and Properties of Matter (H) Student Material 14

31 Examples 1. In a snooker game, the cue ball, of mass 0.2 kg, is accelerated from the rest to a velocity of 2 m s -1 by a force from the cue which lasts 50 ms. What size of force is exerted by the cue? u = 0 v = 2 m s -1 t = 50 ms = 0.05 s m = 0.2 kg F =? Ft = mv - mu F 0.05 = F = 8 N 2. A tennis ball of mass 100 g, initially at rest, is hit by a racket. The racket is in contact with the ball for 20 ms and the force of contact varies over this period as shown in the graph. Determine the speed of the ball as it leaves the racket. Impulse = Area under graph = = 4 N s u = 0 m = 100 g = 0.1 kg v =? Ft = mv - mu = 0.1v 4 = 0.1 v v = 40 m s A tennis ball of mass 0.1 kg travelling horizontally at 10 m s -1 is struck in the opposite direction by a tennis racket. The tennis ball rebounds horizontally at 15 m s -1 and is in contact with the racket for 50 ms. Calculate the force exerted on the ball by the racket. m = 0.1 kg u = 10 m s -1 v = -15 m s -1 (opposite direction to u) t = 50 ms = 0.05 s Ft = mv - mu 0.05 F = 0.1 (-15) = = -2.5 F = = -50 N (Negative indicates force in opposite direction to initial velocity) 0.05 Physics: Mechanics and Properties of Matter (H) Student Material 15

32 Newton s 3rd Law and Momentum Newton s 3rd law states that if an object A exerts a force (ACTION) on object B then object B will exert an equal and opposite force (REACTION) on object A. This law can be proved using the conservation of momentum. Consider a jet engine expelling gases in an aircraft. Let F A be the force on the aircraft by the gases and F G be the force on the gases by the engine (aircraft). Let the positive direction be to the left (direction of Fa) In a small time, let m G be the mass of the gas expelled and m A be the mass of the aircraft. total momentum before = total momentum after O = m G v G + m A v A m G v G = - m A v A [v B and v A in opposite directions] (m G v G - O) = -(m A v A - O) Change in momentum of gas = - (change in momentum of aircraft) Changes in momentum of each object are equal in size but opposite in direction. If forces act in time, t Force = change in momentum t F G = (m G v G - O) FA = (m A v A - O) (u A = u G = O) t t But (m G v G - O) = (m A v A - O) from above F G = - F A since t is the same for the engine and gas. The forces acting are equal in size and opposite in direction. Physics: Mechanics and Properties of Matter (H) Student Material 16

33 DENSITY AND PRESSURE Density The mass per unit volume of a substance is called the density, r. (The symbol,, is the Greek letter rho). = density in kilograms per cubic metre, kg m - 3 m = V m = mass in kilograms, kg V = volume in cubic metres, m 3 Example Calculate the density of a 10 kg block of carbon measuring 10 cm by 20 cm by 25 cm. First, calculate volume, V, in m 3 : V = = m 3 =? = m 10 m = 10 kg V V = m 3 = 2000 kg m -3 Densities of Solids, Liquids and Gases From the table opposite, it can be seen that the relative magnitude of the densities of solids and liquids are similar but the relative magnitude of gases are smaller by a factor of When a solid melts to a liquid, there is little relative change in volume due to expansion. The densities of liquids and solids have similar magnitudes. When a liquid evaporates to a gas, there is a large relative change in volume due to the expansion of the material. The volume of a gas is approximately 1000 times greater than the volume of the same mass of the solid or liquid form of the substance. The densities of gases are smaller than the densities of solids and liquids by a factor of approximately Substance Density (kg m -3 ) Ice 920 Water 1000 Steam 0.9 Aluminium 2700 Iron 7860 Perspex 1190 Ethanol 791 Olive oil 915 Vinegar 1050 Oxygen 1.43 Nitrogen 1.25 It follows, therefore, that the spacing of the particles is a gas must be approximately 10 times greater than in a liquid or solid. Physics: Mechanics and Properties of Matter (H) Student Material 17

34 Pressure Pressure on a surface is defined as the force acting normal (perpendicular) to the surface. p = F A p = pressure in pascals, Pa F = normal force in newtons, N A = area in square metres, m 2 1 pascal is equivalent to 1 newton per square metre; ie 1 Pa = 1 N m -2. Example Calculate the pressure exerted on the ground by a truck of mass 1600 kg if each wheel has an area of 0.02 m 2 in contact with the ground. Total area A = = 0.08 m 2 Normal force F = weight of truck = mg = = N p =? F = N p = F = A = 0.08 m2 A 0.08 Pressure In Fluids = 196,000 Pa or 196 kpa Fluid is a general term which describes liquids and gases. Any equations that apply to liquids at rest equally apply to gases at rest. The pressure at a point in a fluid at rest of density, depth h below the surface, is given by p = h g p = pressure in pascals, Pa h = depth in metres, m = density of the fluid in kg m -3 g = gravitational field strength in N kg -1 Example Calculate the pressure due to the water at a depth of 15 m in water. p =? p = h g h = 15 m = water = 1000 kg m -3 = Pa g = 9.8 N kg-1 Physics: Mechanics and Properties of Matter (H) Student Material 18

35 Buoyancy Force (Upthrust) When a body is immersed in a fluid, it appears to lose weight. The body experiences an upwards force due to being immersed in the fluid. This upwards force is called an upthrust. This upthrust or buoyancy force can be explained in terms of the forces acting on the body due to the pressure acting on each of the surfaces of the body. Pressure on the top surface p top = h top g Pressure on bottom surface p bottom = h bottom g The bottom surface of the body is at a greater depth than the top surface, therefore the pressure on the bottom surface is greater than on the top surface. This results in a net force upwards on the body due to the liquid. This upward force is called the upthrust. Notice that the buoyancy force (upthrust) on an object depends on the difference in the pressure on the top and bottom of the object. Hence the value of this buoyance force does not depend on the depth of the object under the surface. GAS LAWS Kinetic Theory of Gases The kinetic theory tries to explain the behaviour of gases using a model. The model considers a gas to be composed of a large number of very small particles which are far apart and which move randomly at high speeds, colliding elastically with everything they meet. Volume The volume of a gas is taken as the volume of the container. The volume occupied by the gas particles themselves is considered so small as to be negligible. Temperature The temperature of a gas depends on the kinetic energy of the gas particles. The faster the particles move, the greater their kinetic energy and the higher the temperature. Pressure The pressure of a gas is caused by the particles colliding with the walls of the container. The more frequent these collisions or the more violent these collisions, the greater will be the pressure. Physics: Mechanics and Properties of Matter (H) Student Material 19

36 Relationship Between Pressure and Volume of a Gas Summary Notes For a fixed mass of gas at a constant temperature, the pressure of a gas is inversely proportional to its volume. Graph p 1 p V = constant p 1 V 1 = p 2 V 2 V 0 0 Example The pressure of a gas enclosed in a cylinder by a piston changes from 80 kpa to 200 kpa. If there is no change in temperature and the initial volume was 25 litres, calculate the new volume. p 1 = 80 kpa p 1 V 1 = p 2 V 2 V 1 = 25 litres = 200 V 2 p 2 = 200 kpa V 2 = 10 litres V 2 =? Relationship Between Pressure and Temperature of a Gas If a graph is drawn of pressure against temperature in degrees celsius for a fixed mass of gas at a constant volume, the graph is a straight line which does not pass through the origin. When the graph is extended until the pressure reaches zero, it crosses the temperature axis at -273 o C. This is true for all gases. Kelvin Temperature Scale -273 o C is called absolute zero and is the zero on the kelvin temperature scale. At a temperature of absolute zero, 0 K, all particle motion stops and this is therefore the lowest possible temperature. One division on the kelvin temperature scale is the same size as one division on the celsius temperature scale, i.e. temperature differences are the same in kelvin as in degrees celsius, e.g. a temperature increase of 10 C is the same as a temperature increase of 10 K. Note the unit of the kelvin scale is the kelvin, K, not degrees kelvin, K! Physics: Mechanics and Properties of Matter (H) Student Material 20

37 Converting Temperatures Between C and K Summary Notes Converting C to K add 273 Converting K to C subtract 273 If the graph of pressure against temperature is drawn using the kelvin temperature scale, zero on the graph is the zero on the kelvin temperature scale and the graph now goes through the origin. For a fixed mass of gas at a constant volume, the pressure of a gas is directly proportional to its temperature measured in kelvin (K). p T p T = constant p1 T 1 = p T 2 2 Example Hydrogen in a sealed container at 27 C has a pressure of Pa. If it is heated to a temperature of 77 C, what will be its new pressure? p 1 = Pa T 1 = 27 C = 300 K p 2 =? T 2 = 77 C = 350 K p 2 = Pa Physics: Mechanics and Properties of Matter (H) Student Material 21

38 Summary Notes Relationship Between Volume and Temperature of a Gas If a graph is drawn of volume against temperature, in degrees celsius, for a fixed mass of gas at a constant pressure, the graph is a straight line which does not pass through the origin. When the graph is extended until the volume reaches zero, again it crosses the temperature axis at -273 C. This is true for all gases. If the graph of volume against temperature is drawn using the kelvin temperature scale, the graph now goes through the origin. For a fixed mass of gas at a constant pressure, the volume of a gas is directly proportional to its temperature measured in kelvin (K). V T V T = constant V T 1 1 = V 2 T2 Example 400 cm 3 of air is at a temperature of 20 C. At what temperature will the volume be 500 cm 3 if the air pressure does not change? V 1 = 400 cm 3 T 1 = 20 C = 293 K V 2 = 500 cm 3 T 2 =? V T 1 1 = V T = 500 T T 2 = 366 K = 93 C (convert back to temperature scale in the question) Combined Gas Equation By combining the above three relationships, the following relationship for the pressure, volume and temperature of a fixed mass of gas is true for all gases. p V = constant 1 p V 1 1 T = p V T 2 Physics: Mechanics and Properties of Matter (H) Student Material 22

39 Summary Notes Example A balloon contains 1.5 m 3 of helium at a pressure of 100 kpa and at a temperature of 27 C. If the pressure is increased to 250 kpa at a temperature of 127 C, calculate the new volume of the balloon. p 1 = 100 kpa V 1 = 1.5 m V T 1 = 27 C = 300 K = p 2 = 250 kpa V 2 =? V 2 = 0.8 m 3 T 2 = 127 C = 400 K 2 Physics: Mechanics and Properties of Matter (H) Student Material 23

40 Gas Laws and the Kinetic Theory of Gases Summary Notes Pressure - Volume (constant mass and temperature) Consider a volume V of gas at a pressure p. If the volume of the container is reduced without a change in temperature, the particles of the gas will hit the walls of the container more often (but not any harder as their average kinetic energy has not changed). This will produce a larger force on the container walls. The area of the container walls will also reduce with reduced volume. As volume decreases, then the force increases and area decreases resulting in, from the definition of pressure, an increase in pressure, i.e. volume decreases hence pressure increases and vice versa. Pressure - Temperature (constant mass and volume) Consider a gas at a pressure p and temperature T. If the temperature of the gas is increased, the kinetic energy and hence speed of the particles of the gas increases. The particles collide with the container walls more violently and more often. This will produce a larger force on the container walls. As temperature increases, then the force increases resulting in, from the definition of pressure, an increase in pressure, i.e. temperature increases hence pressure increases and vice versa. Volume - Temperature (constant mass and pressure) Consider a volume V of gas at a temperature T. If the temperature of the gas is increased, the kinetic energy and hence speed of the particles of the gas increases. If the volume was to remain constant, an increase in pressure would result as explained above. If the pressure is to remain constant, then the volume of the gas must increase to increase the area of the container walls that the increased force is acting on, i.e. volume decreases hence pressure increases and vice versa. Physics: Mechanics and Properties of Matter (H) Student Material 24

41 ACTIVITY 1 Activities Title: Acceleration Aim: To calculate the acceleration of a trolley moving down a slope. Apparatus: 2 light gates, 1 trolley, 1 slope, 3 stopcocks, 2 power supplies. Instructions Set up the apparatus as shown. Release the trolley from the top of the slope. When clock 1 starts, start clock 3 manually. When clock 2 starts, stop clock 3 manually. Repeat 5 times, ensuring the trolley takes the same path each time. Measure the length of the card. For each run calculate the acceleration. Find the mean acceleration and estimate the random uncertainty. Present your results in table form. Suggest how the experiment could be improved. Run t 1 (s) u (m s -1 ) t 2 (s) v (m s -1 ) t 3 (s) a (m s -2 ) Mean a (m s -2 ) Physics: Mechanics and Properties of Matter (H) Student Material 25

42 ACTIVITY 2A Activities Title: Acceleration Aim: To measure the acceleration of a trolley moving down a slope using a computer. Apparatus: 1 slope, 1 trolley and double mask, 1 light gate, computer and interface, (QED) Apparatus: 1 power supply. Instructions Set up the apparatus as shown in the diagram. After selecting the acceleration program, allow the trolley to run down the track. Note the value of the acceleration. Repeat 5 times. Calculate the mean acceleration and random uncertainty. Explain, in detail, how the mask arrangement allows the computation of the acceleration. ACTIVITY 2B Title: Acceleration (Outcome 3) Apparatus: as in Activity 2A Instructions For 5 different angles of slope find the corresponding acceleration. Using an appropriate format to find the relationship between the angle of slope and the acceleration. ACTIVITY 3 Title: Acceleration Aim: To measure the acceleration due to gravity. Apparatus: 1 light gate, 1 power supply, 1 metal mask, 1 computer and interface. Instructions Set up the apparatus as shown in the diagram. Using the acceleration program, drop the mask, so it cuts the light beam. Repeat 5 times. Calculate the mean value of the acceleration and the random uncertainty. Suggest any improvements to the experiment Physics: Mechanics and Properties of Matter (H) Student Material 26

43 ACTIVITY 4 Activities Title: Resultant of two forces Aim: To compare the resultant of two forces exerted by two tugs on an oil rig with the single Aim: force which produces the same effect on the oil rig. Apparatus: 2 Newton balances, elastic bands, one wooden board, white paper and drawing Apparatus: pins. Instructions Set up the apparatus as shown. With elastic pulled to point P, note the values of F 1, F 2 and trace their direction on the paper below. Mark point P. Using one spring balance, pull the elastic until point P is reached. Note the value required F T. Now combine F 1 and F 2 and compare this with the value of F T. Physics: Mechanics and Properties of Matter (H) Student Material 27

44 ACTIVITY 5 Activities Title: Velocity-time graphs (acceleration-time graphs) Aim: To obtain the velocity - time graph for a) a ball thrown upwards and caught b) a ball dropped continuing to bounce (If a Pasco interface is available an acceleration-time graph can also be obtained). Apparatus: computer, (Pasco) interface, motion sensor, football. Instructions a) Set up the interface to plot a velocity-time (and acceleration-time graph). Throw the ball upwards towards the motion sensor and catch it on the way down. Print the velocity - time graph obtained. b) Repeat only this time allow the ball to bounce on the floor beneath the motion-sensor until it comes to rest. Print the graph(s) obtained. On each graph label the following positions: a) the highest point of the ball b) motion upwards c) motion downwards. Physics: Mechanics and Properties of Matter (H) Student Material 28

45 ACTIVITY 6 Activities Title: Velocity-time graphs (acceleration-time graphs) Aim: To obtain the velocity-time (and acceleration-time) graph for: a) a trolley pushed up a slope and allowed to roll back down b) a trolley released, bouncing against a buffer at the bottom. a) (If a Pasco interface is available an acceleration-time graph can also be obtained). Instructions Set up the apparatus as shown above. Practice pushing the trolley up the slope so it reaches no nearer than 40 cm to the motion sensor. (This is the minimum distance the sensor will register). Stop the trolley when it returns to its starting position. Set up the interface to plot the velocity-time (and acceleration-time) graph. Now use the computer to record the data. Print the graph(s) obtained. Mark on the graphs the relative position of the trolley on the track. b) Set up the apparatus as shown above this time using an explosive trolley. With the same computer set up, obtain the graphs for the trolley when released and allowed to collide with the buffer at the bottom. Allow the trolley to come to rest. Again mark on the graphs the relative position of the trolley on the track. Clearly show the points of collision. Physics: Mechanics and Properties of Matter (H) Student Material 29

46 ACTIVITY 7 Activities Title: Equations of Motion Aim: To calculate the acceleration due to gravity using s = ut at2. Apparatus: electromagnet/trap door, venner stop clock, metre stick, ball bearing. Instructions Measure the distance between the bottom of the ball bearing and the trap door. Allow the ball to fall several times (e.g. 5) and find the average time it takes to reach the trap door. Show that the equation s = ut at2 in this case reduces to h = 1 2 gt2 where h = height dropped, g = acceleration due to gravity Use this equation to calculate the acceleration due to gravity. Estimate the uncertainty in your answer. How could the method be improved? Physics: Mechanics and Properties of Matter (H) Student Material 30

47 ACTIVITY 8 Activities Title: Projectiles Aim: To find the horizontal speed of a projectile. Apparatus: projectile launcher, metre stick, carbon paper, sheet of paper, marble or ball bearing. Instructions Measure the height the ball travels after it leaves the launcher and hits the floor. Use s = ut at2 to calculate the time of flight (remember u v = 0 at the top). Launch the ball and find where it strikes the floor. Place white paper and carbon paper as shown in this area. Measure the horizontal distance travelled (range). Use your results to calculate the horizontal speed. Repeat this 5 times and find the mean value of the horizontal speed. (Ensure ball starts from same point each time). Estimate the random uncertainty. Physics: Mechanics and Properties of Matter (H) Student Material 31

48 ACTIVITY 9 Activities Title: Lifts Aim: To calculate the acceleration of a lift. Apparatus: 1 set of bathroom scales Instructions Use the scales to determine your weight in newtons (N) and your mass in kilograms (kg). Stand on the scales inside the lift and note the reading on the scales while the lift is stationary. Press the button to send the lift upwards. Note: a) the maximum reading on the scales as the lift accelerates upwards b) the reading on the scales as the lift is travelling at a constant speed c) the minimum reading on the scales as the lift decelerates to a halt. Repeat the experiment for the downwards journey of the lift. Use the results to calculate the acceleration of the lift at the three stages of each journey. Physics: Mechanics and Properties of Matter (H) Student Material 32

49 ACTIVITY 10 Activities Title: Work done Aim: To calculate the work done by the force of friction acting on a trolley moving down a slope. Apparatus: 1 light gate & timer, 1 trolley with card, 1 runway / slope, 1 metre stick, scales. Instructions Measure the mass of the trolley, m. Set up the apparatus as shown, marking the initial position of the centre of the trolley on the slope. Measure the distance, d, travelled by the trolley to the light gate. Release the trolley, allowing it to run through the light beam. Note the time recorded on the timer at the light gate and calculate the speed of the trolley, v, through the light gate. Measure the difference in height between the starting position and the position of the light gate; height h. Calculate the loss in gravitational potential energy of the trolley. Calculate the gain in kinetic energy of the trolley. Hence calculate the energy difference due to friction. By considering the work done by friction on the trolley, calculate the average force of friction acting on the trolley over the distance, d, down the runway. Extension By considering the uncertainties in the measured quantities in the above experiment, estimate a value for the absolute uncertainty in the average force of friction for the journey. Physics: Mechanics and Properties of Matter (H) Student Material 33

50 ACTIVITY 11 Activities Title: Momentum Aim: To compare the total momentum before and after a collision. Apparatus: 1 linear air track, 2 vehicles (1 with card), 2 light gates, 1 computer & velocity Apparatus: software, scales. Instructions Note the masses of vehicle A and vehicle B. Set the computer to measure TWO velocities. Keeping vehicle B stationary, push A towards B. Note: a) the velocity of A through light gate 1 b) the velocity of A and B together through light gate 2. Calculate the total momentum before and after the collision. Repeat the experiment for different masses of vehicle A and vehicle B. Use an appropriate format to compare the total momentum before and after the collision. ACTIVITY 12 Title: Conservation of momentum Aim: To compare the total momentum before and after a collision. Apparatus: 1 linear air track, 2 vehicles with cards, 2 light gates, 1 computer & velocity software, scales. Instructions Note the masses of vehicle A and vehicle B. Set the computer to measure 3 velocities. Keeping vehicle B stationary, push A towards B. Note: a) the velocity of A through light gate 1 b) the velocity of B through light gate 2 c) the velocity of A through light gate 1 or 2 (if at all!). Compare the total momentum before and after the collision. Repeat for different masses. Physics: Mechanics and Properties of Matter (H) Student Material 34

51 ACTIVITY 13 Activities Title: Explosions Aim: To compare the total momentum before and after a collision using explosive trolleys. Apparatus: 2 trolleys with cards, 2 light gates, 1 computer & velocity software, scales. Instructions Note the masses of trolleys A and B. Set the computer to measure two velocities. With both trolleys stationary, strike the plunger. Note: a) the velocity of A through light gate 1 b) the velocity of B through light gate 2. Compare the total momentum before and after the explosion. (Remember momentum is a vector quantity). Repeat the experiment for different masses of trolleys. ACTIVITY 14 Title: Impulse Aim: To calculate average force exerted by a putter on a golf ball. Apparatus: 1 putter and mountings, 1 metal painted golf ball with metal tee, 1 light Apparatus: gate, 2 millisecond timers, scales. Instructions Measure the mass of the golf ball, m, and the diameter of the golf ball. Reset both millisecond timers to zero. Place the light gate in front of the ball so that it will pass through the centre of the beam after having been struck by the putter head. Pull the putter head back and allow it to strike the ball. Note the time of contact on timer 1. Using the time recorded on millisecond timer 2, calculate the velocity, v, of the golf ball after being struck by the putter head. Use the equation Ft = mv - mu to calculate the force acting on the ball. Physics: Mechanics and Properties of Matter (H) Student Material 35

52 ACTIVITY 15 Activities Title: Impulse Aim: To compare different times of contact for different balls bouncing on a hard surface. Apparatus: 3 balls of different types, covered in metal foil, 1 retort stand,1 millisecond Apparatus: timer. Instructions Reset the millisecond timer to zero. Raise the first ball to a set height h, e.g. 50 cm, and allow it to bounce once only onto the retort stand base. Note down the time, t, recorded on the millisecond timer. This is the time of contact between the ball and the retort stand base during the bounce. Repeat 3 or 4 times from the same height and obtain an average value for the time. Repeat the experiment for the other two balls, dropping them from the same height. Below are three examples of force-time graphs, showing force varying with time. From the results of your experiment, match the ball used to the graph that could show the variation of force with time during the bounce. By considering the materials the balls are made from, explain the results. Physics: Mechanics and Properties of Matter (H) Student Material 36

53 ACTIVITY 16A Activities Title: Impulse Aim: To compare force-time graphs for different collisions. Apparatus: Pasco interface, motion sensor, force sensor, track, bumper, computer. Instructions Set up the computer to plot graphs of a) force against time b) velocity against time. Give the trolley a gentle push so it collides with the bumper. Use the statistics function to calculate the area under the force - time graph for the first collision. Read off the velocity before and after the collision. Measure the mass of the trolley and hence calculate the change of momentum. Compare this with the impulse calculated (area under the graph). Account for any difference. Repeat the experiment with a slightly stronger push. Explain any differences in the graphs produced. Now replace the spring with clay and repeat the process. Explain the change in the shape of the force-time graph. ACTIVITY 16B Repeat Activity 16A using repelling magnets between the trolley and bumper. Physics: Mechanics and Properties of Matter (H) Student Material 37

54 ACTIVITY 17A Activities Title: Densities of Solids and Liquids Aim: To measure and compare the densities of solid and liquid substances. Apparatus: selection of regular solid shapes, selection of liquids, digital balance, measuring cylinder. Instructions For each solid shape, measure the mass of the solid shape. Measure the dimensions of the solid and calculate its volume. Calculate the density of each of the solids. Present your results in table form. Place the measuring cylinder on the balance and zero the balance. For each liquid, pour the liquid into the measuring cylinder and record its volume and mass. Calculate the density of the liquid. Present your results in table form. Compare your results for the densities of solids and liquids. State any conclusion that can be drawn. ACTIVITY 17B Demonstration Title: Density of Air Aim: To measure the density of air. Apparatus: plastic container with stopper, clamp and tubing, digital balance, pump, basin, Apparatus: measuring cylinder. Instructions Measure the mass of the empty plastic container. Pump air into the container. Measure the mass of the inflated container. Calculate the mass of the air added to the container. Place the end of the tubing under water beneath an upturned, water-filled measuring cylinder. Slowly open the clip on the tubing and measure the volume of air that enters the measuring cylinder. Calculate the density of air using your results. State any sources of inaccuracy that appear during the experiment. State any safety precautions that have to be taken whilst carrying out this experiment. Physics: Mechanics and Properties of Matter (H) Student Material 38

55 ACTIVITY 18A Activities Title: Pressure And Depth In Fluids (Outcome 3) Apparatus: pressure sensor, rubber tubing connected to inverted thistle funnel with rubber diaphragm (can use a balloon), meter and tall measuring cylinder filled with a liquid. Instructions Record the pressure reading at different depths in the same liquid. Use an appropriate format to show the relationship of pressure due to a fluid and depth. ACTIVITY 18B Title: Pressure And Density In Fluids (Outcome 3) Apparatus: pressure sensor, rubber tubing connected to hollow glass tube, meter and Apparatus: measuring cylinders filled with different liquids. Instructions Record the pressure reading at the same depth in different liquids. Use an appropriate format to show the relationship of pressure due to a fluid and density. Physics: Mechanics and Properties of Matter (H) Student Material 39

56 ACTIVITY 19A Activities Title: Relationship between pressure and volume of a fixed mass of gas (Outcome 3) at a fixed temperature. Apparatus: Boyle s Law apparatus, pump. Instructions Use the pump to increase the pressure on the column of trapped air. Seal the apparatus using the tap when the pressure is high. Record the length of the trapped air column and the corresponding pressure. Using the tap, slowly reduce the pressure on the oil and seal the apparatus at a ne value of length. Record the new value of length and corresponding pressure. Repeat for a range of values of length of trapped air column. Use an appropriate format to show the relationship between pressure and volume of a gas at constant temperature. ACTIVITY 19B Title: Relationship between pressure and volume of a fixed mass of gas (Outcome 3) at a fixed temperature. Apparatus: gas syringe, rubber tubing, pressure sensor, computer interface. Instructions Set the volume of air to its maximum and record the corresponding pressure. Repeat the process by gently compressing the air in the syringe to obtain a set of readings of volume against corresponding pressure. Use an appropriate format to show the relationship between pressure and volume of a gas at constant temperature. Physics: Mechanics and Properties of Matter (H) Student Material 40

57 ACTIVITY 20A Activities Title: Relationship between pressure and temperature of a fixed mass of (Outcome 3) gas at a fixed volume. Apparatus: air-filled flask, thermometer and glass tube bored through a rubber bung, rubber tubing, bourdon gauge, container of boiling water. Instructions Set up the apparatus as shown and carefully heat the water to boiling. Record the temperature and corresponding pressure. Remove the flask and allow to cool in the air of the room. As the air in the flask cools, obtain a set of corresponding values of temperature and pressure for the air in the flask. Use an appropriate format to show the relationship between pressure and temperature of a gas at constant volume. ACTIVITY 20B Title: Relationship between pressure and temperature of a fixed mass of (Outcome 3) gas at a fixed volume. Apparatus: air-filled flask, temperature probe and glass tube bored through a rubber bung, Apparatus: rubber tubing, pressure sensor, computer interface, container of boiling water. Instructions Set up the apparatus above, connecting the temperature probe to either a digital thermometer or into the computer interface. The rest of the method is as above in Activity 20A. Physics: Mechanics and Properties of Matter (H) Student Material 41

58 ACTIVITY 21 Activities Title: Relationship between volume and temperature of a fixed mass of (Outcome 3) gas at a fixed pressure. Apparatus: capillary tube with mercury plug, scale, thermometer, large beaker of water, Apparatus: bunsen burner. Instructions Set up the apparatus as shown and carefully heat the water to boiling. Record the temperature and corresponding length of the trapped air column. Remove the heat from under the beaker and allow the apparatus to cool. As the column of air cools, obtain a set of corresponding values of temperature and length of trapped air column. Use an appropriate format to show the relationship between volume and temperature of a gas at constant pressure. Physics: Mechanics and Properties of Matter (H) Student Material 42

59 Answers REVISION PROBLEMS Speed 1. The world downhill speed skiing trial takes place at Les Arc every year. Describe a method that could be used to find the average speed of the skier over the 1km run. Your description should include: a) any apparatus required b) details of what measurements need to be taken c) an explanation of how you would use the measurements to carry out the d) calculations. 2. An athlete ran a 1500 metres race in 3 minutes 40 seconds. Find his average speed for the race. 3. How far away is the sun if it takes light 8 minutes to reach Earth? (Speed of light = m s -1 ). 4. Concorde travels at an average speed of Mach 1.3 between London and New York. Calculate the time for the journey to the nearest minute. The distance between London and New York is 4800 km. (Mach 1 is the speed of sound. Take the speed of sound to be 340 m s -1 ). 5. The speed - time graph below represents a girl running for a bus. She starts from a standstill at O and jumps on the bus at Q. v/m s -1 t / s Find: a) the steady speed at which she runs b) the distance she runs c) the increase in the speed of the bus while the girl is on it d) how far the bus travels during QR e) how far this girl travels during OR. 6. A ground-to-air guided missile accelerates from rest at 150 m s -2 for 5 seconds. What speed does it reach? 7. An Aston Martin accelerated from rest at 6 m s -2. How long does it take to reach a speed of 30 m s -1? 8. If a family car applies its brakes when travelling at its top speed of 68 m s -1, and decelerates at 17 m s -2, how long does it take to reduce its speed 34 m s -1? Physics: Mechanics and Properties of Matter (H) Student Material 43

60 Answers Acceleration 9. An armour-piercing shell, travelling at 2000 m s -1, buries itself in the concrete wall of a bunker. If it decelerates at m s -2, what time does it take to come to rest after striking the wall? 10. A skateboard running from rest down a concrete path of uniform slope reaches a speed of 8 m s -1 in 4 s. What is the acceleration of the skateboard? How long after it started would the skateboard take to reach a speed of 12 m s -1? 11. In the Tour de France a cyclist is travelling at 20 m s -1. When he reaches a downhill stretch his speed increases to 40 m s -1. It takes 4 s for him to reach this point on the hill. What is the acceleration of the cyclist on the hill? Assuming he maintains this acceleration, how fast will he be travelling after a further 2 s? How long would it take the cyclist to reach a speed of 55 m s -1? 12. Use the information given below to calculate the acceleration of the trolley. Length of card = 5 cm Time on clock 1 = 0.10 s (time taken for card to interrupt top light gate) Time on clock 2 = 0.05 s (time taken for card to interrupt bottom light gate) Time on clock 3 = 2.50 s (time taken for trolley to travel between top and bottom light gate) 13. A pupil uses light gates and a suitably interfaced computer to measure the acceleration of a trolley as it moves down an inclined plane. The following results were obtained: acceleration (m s -2 ) 5.16, 5.24, 5.21, 5.19, 5.20, 5.20, 5.17, Calculate the mean valve of the acceleration and the corresponding random uncertainty. Physics: Mechanics and Properties of Matter (H) Student Material 44

61 Answers MECHANICS AND PROPERTIES OF MATTER PROBLEMS Vectors 14. A car travels 50 km N and then returns 30 km S. The whole journey takes 2 hours. Calculate: a) the distance travelled Calculate: b) the average speed Calculate: c) the displacement Calculate: d) the average velocity. 15. A girl delivers newspapers to three houses,, Y, Z, as shown in the diagram, starting at. The girl walks directly from one house to the next. Calculate the total distance the girl walks. Calculate the girl s final displacement from. If the girl walks at a steady speed of 1 m s -1, calculate the time she takes to get from to Z. Calculate her resultant velocity. 40 m 16. Find the resultant force in the following cases: 30 m 17. An aircraft has a maximum speed of 1000 km h -1. If it is flying north into a headwind speed 100 km h -1 what is the maximum velocity of the aircraft? 18. A model aircraft is flying north with a velocity of 24 m s -1. A wind is blowing from west to east at 10 m s -1. What is the resultant velocity of the plane? 19. An aircraft pilot wishes to fly north at 800 km h -1. A wind is blowing at 80 km h -1 from west to east. What speed and course must he select in order to fly the desired course? 20. State what is meant by a vector quantity and scalar quantity. Give two examples of each. 21. Find the average speed and average velocity of the following. An orienteer who runs 5 km due South, 4 km due West and then 2 km North in 1 hour. 22. A ship is sailing East at 4 m s -1. A passenger walks due North at 2 m s -1. What is the resultant velocity of the passenger relative to the sea? Physics: Mechanics and Properties of Matter (H) Student Material 45

62 Answers (Use both scale drawing and trigonometry). 23. A man pulls a garden roller with a maximum force of 50 N. Find his effective horizontal force. Without changing the force applied, explain how he could increase this effective force. 24. A barge is dragged along a canal as shown below. barge What is the component of the force parallel to the canal? 25. A toy train of mass 0.2 kg, is given a push of 10 N at an angle of 30 0 to the rails. Calculate a) the component of force along the rails b) the acceleration of the train. 26. A football is kicked up at an angle of 70 0 at 15 m s -1. Calculate a) the horizontal component of the velocity b) the vertical component of the velocity? Physics: Mechanics and Properties of Matter (H) Student Material 46

63 Answers Equations of Motion 27. The graph below shows how the acceleration of an object varies with time. The object started from rest. a / m s t / s Draw a velocity time graph for the first 10 s of the motion. 28. The velocity time graph for an object is shown below. v / m s -1 Velocity 10 (m s -1 ) 5 t / s Draw the corresponding acceleration-time graph. (Put numerical values v on / m time s -1 axis) The graph shows the velocity of a ball which is dropped and bounces from a floor. A downwards direction is taken as being positive B E t / s 0 - C D During section OB of the graph in which direction is the ball travelling? what can you say about the speed of the ball? During section CD of the graph in which direction is the ball travelling? what can you say about the speed of the ball? During section DE of the graph in which direction is the ball travelling? what can you say about the speed of the ball? What happened to the ball at point B on the graph? What happened to the ball at point C on the graph? What happened to the ball at point D on the graph? How does the speed of the ball immediately after rebound compare with the speed immediately before? Physics: Mechanics and Properties of Matter (H) Student Material 47

64 30. Which velocity-time graph below represents the motion of a ball which is thrown vertically upwards and returns to the thrower 3 seconds later? Answers 31. A ball is dropped from a height and bounces up and down on a horizontal surface. Assuming that there is no loss of kinetic energy at each bounce, select the velocity-time graph which represents the motion of the ball from the moment it is released. v / m s -1 v / m s -1 v / m s -1 v / m s -1 v / m s -1 v / m s -1 t / s t / s t / s A B t / s A B v / m s -1 v / m s -1 v / m s -1 v / m s t / s t / s D E D E C C t / s t / s t / s t / s 32. A ball is dropped from rest and bounces several times, losing some kinetic energy at each bounce. Selected the correct velocity - time graph for this motion. v / m s -1 v / m s -1 v / m s -1 t / s t / s t / s A B C v / m s -1 t / s v / m s -1 t / s D E Physics: Mechanics and Properties of Matter (H) Student Material 48

65 Answers 33. An object accelerates uniformly at 4 m s -2 from an initial speed of 8 m s -1. How far does it travel in 10 s? 34. A car accelerates uniformly at 6 m s -2, its initial speed is 15 m s -1 and it covers a distance of 200 m. Calculate its final velocity. 35. A ball is thrown to a height of 40 m above its starting point, with what velocity was it thrown? 36. A car travelling at 30 m s -1 slows down at 1.8 m s -2 over a distance of 250 m. How long does it take to stop? 37. If a stone is thrown vertically down a well at 5 m s -1. Calculate the time taken for the stone to reach the water surface 60 m below. 38. A tennis ball launcher is 0.6 m long and the velocity of a tennis ball leaving the launcher is 30 m s -1. Calculate: a) the average acceleration of a tennis ball b) the time of transit in the launcher. 39. In an experiment to find g a steel ball falls from rest through 40 cm. The time taken is 0.29 s. What is the value for g. 40. A trolley accelerates down a slope. Two photo-cells spaced 0.5 m apart measure the velocities to be 20 cm s -1 and 50 cm s -1. Calculate a) the acceleration of the trolley b) the time taken to cover the 0.5 m. 41. A helicopter is rising vertically at 10 m s -1 when a wheel falls off. The wheel hits the ground 8 s later. Calculate at what height the helicopter was flying when the wheel came off. 42. A ball is thrown upwards from the side of a cliff as shown below. a) Calculate: i) the height of the ball above sea level after 2 s ii) the ball s velocity after 2 s. b) What is the total distance travelled by the ball from launch to landing in the sea? Physics: Mechanics and Properties of Matter (H) Student Material 49

66 Answers 43. A box is released from a plane travelling with a horizontal velocity of 300 m s -1 and a height of 300 m, find: a) how long it takes the box to hit the ground b) the horizontal distance between the point of impact and the release point c) the position of the plane relative to the box at the time of impact. 44. A projectile is fired horizontally from the edge of a cliff at 12 m s -1 and hits the sea 60 m away. Find: a) the time of flight b) the height of the starting point above sea level. State any assumptions you have made. 45. A ball is projected horizontally at 15 m s -1 from the top of a vertical cliff. It reaches the horizontal ground 45 m from the foot of the cliff. a) Draw graphs, giving appropriate numerical values of the ball s i) horizontal speed against time ii) vertical speed against time, for the period between projection until it hits the ground b) Use a vector diagram, to find the velocity of the ball 2 s after its projection. (Magnitude and direction are required). 46. A projectile is fired across level ground taking 6 s to travel from A to B. The highest point reached is C. Air resistance is negligible. v H /m s -1 v v /m s -1 t / s t / s a) Describe: i) the horizontal motion of the projectile ii) the vertical motion of the projectile? b) Use a vector diagram, to find the speed and angle at which the projectile was fired from point A. c) Find the speed at position C. Explain why this is the smallest speed of the projectile. d) Calculate the height above the ground of point C. e) Find the range AB. Physics: Mechanics and Properties of Matter (H) Student Material 50

67 Answers 47. An object of mass 5 kg is propelled with a speed of 40 m s -1 at an angle of 30 o to the horizontal. Find: a) the vertical component of its initial velocity b) the maximum vertical height reached c) the time of flight for the whole trajectory d) the horizontal range of the object. 48. A missile is launched at 60 to the ground and strikes a target on a hill as shown below. If the initial speed of the missile was 100 m s -1 find: a) the time taken to reach the target b) the height of the target above the launcher. 49. A stunt driver hopes to jump across a canal of width 10 m. The drop to the other side is 2 m as shown. a) Calculate the horizontal speed required to make it to the other side b) State any assumptions you have made. 50. Describe how you could measure the acceleration of trolley starting from rest moving down a slope. You are provided with a metre stick and stop clock. Your description should include: a) a diagram b) a list of measurements taken c) how you would use these measurements to calculate the acceleration. d) how you would estimate the uncertainties involved in the experiment. Physics: Mechanics and Properties of Matter (H) Student Material 51

68 Answers Newton s 2nd Law, energy and power 51. State Newton s 1st Law of Motion. 52. A lift of mass 500 kg travels upwards at a constant speed. Calculate the tension in the lifting table. 53. (a) A fully loaded oil tanker has a mass of kg. As the speed of the tanker increases from zero to a steady maximum speed of 8.0 m s -1 the force from the propellers remains constant at N. (i) (ii) Calculate the acceleration of the tanker just as it starts from rest. What is the size of the force of friction acting on the tanker when it is travelling at the steady speed of 8.0 m s -1? (b) When its engines are stopped, the tanker takes 50 minutes to come to rest from a speed of 8.0 m s -1. Calculate its average deceleration. 54. v / m s -1 A B C D 0 E t / s The graph shows how the speed of a parachutist varies with time after having jumped from the aeroplane. With reference to the letters, explain each stage of the journey. 55. Two girls push a car of mass 2000 kg. Each applies a force of 50 N and the force of friction is 60 N. Calculate the acceleration of the car. 56. A boy on a skateboard rides up a slope. The total mass of the boy and the skateboard is 90 kg. He decelerates uniformly from 12 m s -1 to 2 m s -1 in 6 seconds. Calculate the resultant force acting on him. 57. A box is pulled along a rough surface with a constant force of 140 N. If the mass of the box is 30 kg and it accelerates at 4 m s -2 calculate: (a) the unbalanced force causing the acceleration (b) the force of friction between the box and the surface. Physics: Mechanics and Properties of Matter (H) Student Material 52

69 58. An 800 kg Metro is accelerated from 0 to 18 m s -1 in 12 seconds. (a) What is the resultant force acting on the Metro? (b) How far does the car travel in these 12 seconds? At the end of the 12 s period the brakes are operated and the car comes to rest in a distance of 50 m. (c) What is the average frictional force acting on the car? Answers 59. (a) A rocket of mass kg is launched vertically upwards. Its engines produce a constant thrust of N. (i) Draw a diagram showing all the forces acting on the rocket. (ii) Calculate the initial acceleration of the rocket. (b) As the rocket rises its acceleration is found to increase. Give three reasons for this. (c) Calculate the acceleration of the same rocket from the surface of the Moon if the Moon s gravitational field strength is 1.6 N kg -1. (d) Explain in terms of Newton s laws of motion why a rocket can travel from the Earth to the Moon and for most of the journey not burn up any fuel. 60. A rocket takes off and accelerates to 90 m s -1 in 4 s. The resultant force acting on it is 40 kn upwards. (a) Calculate the mass of the rocket. (b) Calculate the force produced by the rocket s engines if the average force of friction is 5000 N. 61. What is the minimum force required to lift a helicopter of mass 2000 kg upwards with an initial acceleration of 4 m s -2. Air resistance is 1000 N. 62. A crate of mass 200 kg is placed on a balance in a lift. (a) What would be the reading on the balance, in newtons, when the lift was stationary? (b) The lift now accelerates upwards at 1.5 m s -2. What is the new reading on the balance? (c) The lift then travels up at a constant speed of 5 m s -1. What is the reading on the balance? (d) For the last stage of the journey calculate the reading on the balance when the lift decelerates at 1.5 m s -2 while moving up. 63. A small lift in a hotel is fully loaded and has a mass of 250 kg. For safety reasons the tension in the pulling cable must never be greater than 3500 N. (a) What is the tension in the cable when the lift is: (i) at rest (ii) moving up at a constant 1 m s -1 (iii) accelerating upwards at 2 m s -2 (iv) accelerating downwards at 2 m s -2? (b) Calculate the maximum permitted upward acceleration of the fully loaded lift. (c) Describe a situation where the lift could have an upward acceleration greater than the value in (b) without breaching safety regulations. Physics: Mechanics and Properties of Matter (H) Student Material 53

70 Answers 64. A package of mass 4 kg is hung from a spring balance attached to the ceiling of a lift which is accelerating upwards at 3 m s -2. What is the reading on the spring balance? 65. The graph shows how the downward speed of a lift varies with time. v / m s t / s (a) Draw the corresponding acceleration/time graph. (b) A 4 kg mass is suspended from a spring balance inside the lift. Determine the reading on the balance at each stage of the motion. 66. Two masses are pulled along a flat surface as shown below. Find the (a) acceleration of the masses (b) tension, T. 67. A car of mass 1200 kg tows a caravan of mass 1000 kg. The frictional force on the car and caravan is 200 N and 500 N respectively. The car accelerates at 2 m s -2. (a) Calculate the force exerted by the engine of the car. (b) What force does the tow bar exert on the caravan? (c) The car then travels at a constant speed of 10 m s -1. Assuming the frictional forces to be unchanged calculate the new engine force and the force exerted by the tow bar on the caravan. (d) The car brakes and decelerates at 5 m s -2. Calculate the force exerted by the brakes (assume the other frictional forces remain constant). Physics: Mechanics and Properties of Matter (H) Student Material 54

71 Answers 68. A tractor of mass 1200 kg pulls a log of mass 400 kg. The tension in the tow rope is 2000 N and the frictional force on the log is 800 N. How far will the log move in 4 s assuming it was stationary to begin with? 69. A force of 60 N pushes three blocks as shown. If each block has a mass of 8 kg and the force of friction on each block is 4 N calculate: (a) the acceleration of the blocks (b) the force block A exerts on block B (c) the force block B exerts on block C. The pushing force is then reduced until the blocks move at constant speed. (d) Calculate the value of this pushing force. (e) Does the force block A exerts on block B now equal the force block B exerts on block C? Explain. 70. A 2 kg trolley is connected by string to a 1 kg mass as shown. The bench and pulley are frictionless. (a) Calculate the acceleration of the trolley. (b) Calculate the tension in the string. 71. A force of 800 N is applied to a canal barge by means of a rope angled at 40 to the direction of the canal. If the mass of the barge is 1000 kg and the force of friction between the barge and the water is 100 N find the acceleration of the barge. 72. A crate of mass 100 kg is pulled along a rough surface by two ropes at the angles shown. (a) If the crate is moving at a constant speed of 1 m s -1 what is the force of friction? (b) If the forces were increased to 140 N at the same angle calculate the acceleration of the crate. Physics: Mechanics and Properties of Matter (H) Student Material 55

72 Answers 73. A 2 kg block of wood is placed on the slope shown. It remains stationary. What is the size of the frictional force acting up the slope? 74. A 500 g trolley runs down a runway which is 2 m long and raised 30 cm at one end. If its speed remains constant throughout calculate the force of friction acting up the slope. 75. The brakes on a car fail while it is parked at the top of a hill. It runs down the hill for a distance of 50 m until it crashes into a hedge. The mass of the car is 900 kg and the hill makes an angle of 15 to the horizontal. If the average force of friction is 300 N find: (a) the component of weight acting down the slope (b) the acceleration of the car (c) the speed of the car as it hits the hedge (d) the force acting perpendicular to the car (the reaction) when it is on the hill. Momentum and impulse 76. What is the momentum of the object in each of the following situations : (a) (b) (c) 77. A trolley of mass 2 kg and travelling at 1.5 m s -1 collides and sticks to another stationary trolley of mass 2 kg. Calculate the velocity after the collision. Show that the collision is inelastic. 78. A target of mass 4 kg hangs from a tree by a long string. An arrow of mass 100 g is fired with a velocity of 100 m s -1 and embeds itself in the target. At what velocity does the target begin to move after the impact? Physics: Mechanics and Properties of Matter (H) Student Material 56

73 Answers 79. A trolley of mass 2 kg is moving at constant speed when it collides and sticks to a second trolley which was originally stationary. The graph shows how the speed of the 2 kg trolley varies with time. v / m s t / s Determine the mass of the second trolley. 80. In a game of bowls one particular bowl hits the jack straight on causing it to move forward. The jack has a mass of 300 g and was originally stationary, the bowl has a mass of 1 kg and was moving at a speed of 2 m s -1. (a) What is the speed of the jack after the collision if the bowl continued to move forward at 1.2 m s -1? (b) How much kinetic energy is lost during the collision? 81. In space two spaceships make a docking manoeuvre (joining together). One spaceship has a mass of 1500 kg and is moving at 8 m s -1. The second spaceship has a mass of 2000 kg and is approaching from behind at 9 m s -1. Determine their common velocity after docking. 82. Two cars are travelling along a racing track. The car in front has a mass of 1400 kg and is moving at 20 m s -1 while the car behind has a mass of 1000 kg and is moving at 30 m s -1. They collide and the car in front moves off with a speed of 25 m s -1. (a) Determine the speed of the rear car after the collision. (b) Show clearly whether this collision was elastic or inelastic. 83. One vehicle approaches another from behind as shown. The vehicle at the rear is moving faster than the one in front and they collide which causes the vehicle in front to be nudged forward with an increased speed. Determine the speed of the rear vehicle after the collision. 84. A trolley of mass 0.8 kg, travelling at 1.5 m s -1 collides head on with another vehicle of mass 1.2 kg, travelling at 2.0 m s -1 in the opposite direction. They lock together on impact. Determine the speed and direction after the collision. Physics: Mechanics and Properties of Matter (H) Student Material 57

74 Answers 85. A firework is launched vertically and when it reaches its maximum height it explodes into 2 pieces. One piece has a mass of 200 g and moves off with a speed of 10 m s -1. If the other piece has a mass of 120 g what speed does it have? 86. Two trolleys in contact, initially at rest, fly apart when a plunger is released. One trolley with a mass of 2 kg moves off with a speed of 4 m s -1 and the other with a speed, in the opposite direction, of 2 m s -1. What is the mass of this trolley? 87. A man of mass 80 kg and woman of mass 50 kg are skating on ice. At one point they stand next to each other and the woman pushes the man who then moves away at 0.5 m s -1. With what speed and at what direction does the woman move off? 88. Two trolleys in contact, initially at rest, fly apart when a plunger is released. If one has a mass of 2 kg and moves off at speed of 2 m s -1, calculate the velocity of the other trolley given its mass is 3 kg. 89. A cue exerts an average force of 7 N on a stationary snooker ball of mass 200 g. If the impact lasts for 45 ms, with what speed does the ball leave the cue? 90. A girl kicks a football of mass 500 g which was originally stationary. Her foot is in contact with the ball for a time of 50 ms and the ball moves off with a speed of 10 m s -1. Calculate the average force exerted on the ball by her foot. 91. A stationary golf ball is struck by a club. The ball which has a mass of 100 g moves off with a speed of 30 m s -1. If the average force of contact is 100 N calculate the time of contact. 92. The graph shows how the force exerted on a hockey ball by a hockey stick varies with time. If the mass of the ball is 150 g determine the speed of the ball as it leaves the stick (assume that it was stationary to begin with). F/N 40 N 20 t / ms 93. A ball of mass 100 g falls from a height of 20 cm onto a surface and rebounds to a height of 18 cm. The duration of impact is 25 ms. Calculate: (a) the change in momentum of the ball caused by the bounce (b) the average force exerted on the ball by the surface. Physics: Mechanics and Properties of Matter (H) Student Material 58

75 Answers 94. A rubber ball of mass 40 g is dropped from a height of 0.8 m onto the pavement. It rebounds to a maximum height of 0.45 m. The average force of contact between the pavement and the ball is 2.8 N. (a) Calculate the velocity of the ball just before it hits the ground and the velocity just after hitting the ground. (b) Calculate the time of contact between the ball and pavement. 95. A ball of mass 400 g travels horizontally along the ground and collides with a wall. The velocity / time graph below represents the motion of the ball for the first 1.2 seconds. v / m s -1 6 B A C E t / s -4 D (a) Describe the motion of the ball during sections AB, BC, CD and DE. (b) What is the time of contact with the wall? (c) Calculate the average force between the ball and the wall. (d) How much energy is lost due to contact with the wall? 96. Water is ejected from a fire hose at a rate of 25 kg s -1 and a speed of 50 m s -1. If the water hits a wall calculate the average force exerted on the wall. Assume that the water does not rebound from the wall. 97. A rocket burns fuel at a rate of 50 kg per second, ejecting it with a constant speed of 1800 m s -1. Calculate the force exerted on the rocket. 98. Describe in detail an experiment which you would do to determine the average force between a football boot and a football as it is being kicked. Draw a diagram of the apparatus and include all measurements taken and calculations carried out. 99. A 2 kg trolley travelling at 6 m s -1 collides with a stationary 1 kg trolley. (a) If they remain connected, calculate: (i) their combined velocity (ii) the momentum gained by the 1 kg trolley (iii) the momentum lost by the 2 kg trolley. (b) If the collision time is 0.5 s, find the force acting on each trolley. Physics: Mechanics and Properties of Matter (H) Student Material 59

76 Answers Density and pressure 1. A block of iron has a mass of kg and a volume of 5 m 3. What is its density? 2. What is the mass of a cylinder of aluminium with a volume of 2.5 m 3? (The density of aluminium is 2700 kg m -3 ) 3. 1 kg of nitrogen gas is used to fill a balloon. If the density of nitrogen is 1.25 kg m -3, find the volume of the balloon. 4. A tank measures 60 cm long and 40 cm wide. 72 kg of water are to be poured into the tank. How deep will the water be? (Density of water is 1000 kg m -3 ) 5. What will be the mass of air in a classroom with the dimensions: length 15 m, breadth 10 m and height 4 m? (The density of air is 1.3 kg m -3 ). 6. In the diagram below the piston contains 0.2 kg of oxygen, which has a density of 1.43 kg m -3. (a) If the plunger is at height of 40 cm, what is the cross-sectional area of the plunger? (b) The gas is heated and the plunger rises a further 20 cm. What is the density of the oxygen now? m 3 of water is heated until it all changes to steam. What will be the approximate volume of the steam? 8. Solids and liquids are approximately 1000 times denser than gases. How will the separation of the particles compare in each case? 9. Describe an experiment to measure the density of air. Your answer should include: (a) a diagram of the apparatus used (b) a list of measurements taken (c) any necessary calculations. Physics: Mechanics and Properties of Matter (H) Student Material 60

77 10. Explain why the use of large tyres helps to prevent a tractor from sinking into soft ground. Answers 11. A box weighs 120 N and has a base area of 2 m 2. What pressure does it exert on the ground? 12. If atmospheric pressure is Pa, what force does the air exert on a wall of area 10 m 2? 13. A rectangular steel block measures 10 cm 8 cm 6 cm. What is the greatest and the least pressure which it can exert on a surface? (Density of steel is 8000 kg m -3 ) 14. Estimate the pressure you exert on the floor when you are standing on one foot. 15. What is the pressure due to a depth of 10 m of water? (Density of water = 1000 kg m- 3 ). 16. A water tank has a base of cross-sectional area 0.5 m 3 and a depth of water 1.5 m. Calculate: (a) the pressure at the bottom of the tank (caused by the water and by the pressure of the air above the water ) (b) the resultant force on the base. 17. Water is supplied to flats from a tank on the roof. Find the extra water pressure available to residents on the ground floor if their taps are 30 m below the level of the taps on the top floor. 18. A cube of side 12 cm is completely immersed in a liquid of density 800 kg m -3, so that the top surface is horizontal and 20 cm below the surface of the liquid. Calculate the fluid pressure: (a) at a depth of 20 cm (b) at a depth of 30 cm. (c) Hence calculate the force exerted on: i) the top surface of the cube ii) the bottom surface of the cube. (d) Calculate the size and direction of the vertical force due to this pressure difference. Physics: Mechanics and Properties of Matter (H) Student Material 61

78 19. A cube of wood of sides 0.5 m floats in a large tank of water at a height of 0.3 m above the surface. Calculate the: (a) weight of the cube (b) vertical upthrust given to the cube by the water (c) volume of water displaced by the cube. Answers (density of water = 1000 kg m -3 ; density of wood = 400 kg m -3 ) 20. Explain why it is very easy to float in the Dead Sea. Gas laws Pressure and volume (constant temperature) cm 3 of air is contained in a syringe at atmospheric pressure ( 10 5 Pa ). If the volume is reduced to a) 50 cm 3 or b) 20 cm 3 without a change in temperature, what will be the new pressures? 22. If the piston in a cylinder containing 300 cm 3 of gas at a pressure of 10 5 Pa is moved outwards so that the pressure of the gas falls to Pa, find the new volume of the gas. 23. A weather balloon contains 80 m 3 of helium at normal atmospheric pressure of 10 5 Pa. What will be the volume of the balloon at an altitude where air pressure is Pa? 24. The cork in a pop-gun is fired when the pressure reaches 3 atmospheres. If the plunger is 60 cm from the cork when the air in the barrel is at atmospheric pressure, how far will the plunger have to move before the cork pops out? Physics: Mechanics and Properties of Matter (H) Student Material 62