Starters and activities in Mechanics MEI conference 2012 Keele University Starters Centre of mass: two counter-intuitive stable positions of equilibrium The directions of displacement, velocity and acceleration The total surface force on an accelerating car Relative velocity demonstrations and a simulation in GeoGebra Drawing diagrams in mechanics Interpretation of a velocity-time graph Activities An experiment and mathematical model for sliding An experiment and simulation of three forces in equilibrium Simulations to aid understanding of projectile motion A simulation of forces on a box causing sliding or tipping MEI conference 2012 1 of 1
Starters and activities introduction 1. Emily Rae and David Holland 2. These starters and activities are intended to suggest types of things you can do for yourselves to meet the needs of your students. 3. Students have a ready interest in Mechanics because of the way it explains things. This intellectual curiosity is a much better motivator than appeals to utility. 4. Some of Mechanics is quite hard. Students do not give up using their own limited or wrong models just because we give them better ones. They have to be confronted with situations that evidently cannot be correctly analysed by their limited or wrong models. 5. An underlying theme will be the use of simulations as a powerful learning aid. This is a type of aid that has vastly increased in importance with the development of computers.
Centre of mass discussion starters The first activity is a common party trick which is intended to address the issue of whether an object s centre of mass can exist outside of the object itself (for example, in an L shaped object) a concept which students can struggle with when calculating the position of a centre of mass. First, position a fork and spoon end-to-end by pushing the curved part of the spoon through the prongs of the fork. Balance the conjoined cutlery on the end of a matchstick (see image) then balance the other end of the match onto the rim of a glass. The whole system looks improbable, as though it should not be in equilibrium. Ask the students why they think it works, and why the fork and spoon do not fall off the end of the matchstick (the reason is that the centre of mass of the spoon-fork-matchstick system is in a vertical line through the point of contact of the matchstick with the glass) Another discussion starter on the topic of centres of mass is to explore where the centre of mass will be in a can of liquid at various stages of the can emptying. When the can is empty or full, the centre of mass will be in the same place (roughly halfway up the can) However, when a small amount of liquid is removed from the full can the centre of mass is lowered. Ask the students when they think the centre of mass will be at its highest, and when it will be at its lowest. It is also possible to drink just the right amount of liquid to make the can balance on its rim without falling when the can is balanced the centre of mass is located directly above the point where the rim sits on the surface of the table. MEI Conference 2012 1 of 1 20/07/2012
Notes for teachers The direction of displacement, velocity and acceleration Students often make mistakes when asked for the direction of displacements, velocities and accelerations. Common errors are to confuse the direction of motion with the direction of the displacement and, particularly, assume that the direction of the acceleration is the direction of the velocity. Getting a class to consider some sort of periodic motion is very effective. Useful scenarios are walking up and down a line in front of the class, considering a vertical mass-spring oscillator (with a long period), considering the motion of a lift. In each case an origin and the positive direction is defined and for different stages of the motion the class have to decide on the sign of the displacement, velocity and acceleration. It is a good idea to use the walking and lift examples initially as they can be paused to allow discussion. After a longer introductory session, the exercise can be repeated quickly as a starter to reinforce the ideas. Repeating one of the scenarios with a different origin and/or positive direction is very effective. Taking the lift example, one could discuss the non- stop motion from the ground to the 2 nd floor and then the non-stop return journey. Put up a diagram showing the origin at 1 st floor level and the positive direction upwards. Suppose that the lift is initially at ground floor level. Setting off up from ground floor displ ve vel +ve acc +ve Passing 1 st floor displ 0 vel +ve acc 0 (discuss?) Approaching 2 nd floor displ +ve vel +ve acc ve Setting off down from 2 nd floor displ +ve vel ve acc ve Passing 1 st floor displ 0 vel ve acc 0 (discuss?) Approaching ground floor displ ve vel ve acc +ve Now try this with positive downwards. Now try with the lift stopping at 1 st floor level. Now try MEI Conference 2012 1 of 1 20/07/2012
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Total surface force on a car Each of the diagrams shows the total surface contact forces acting on the front and on the rear wheels of a car on a straight horizontal road. Air resistance should be neglected. For each case, suggest a scenario and the motion that could be taking place. Notes for teachers Students may not be familiar with the use of total surface contact force and this may require some discussion. Perhaps a further discussion might be about why we usually find it more convenient to deal with this force in component form, i.e. as friction and normal reaction. Many students will not be familiar with the fact that the frictional force on a car acts towards the front when it is accelerating forwards and this often provokes discussion. In a real situation, the two normal reaction forces will change under acceleration and deceleration while retaining the same total magnitude (equal to the magnitude of the weight of the car). Some students may be aware of this (the change in load on the front and rear suspension is indicated by the car s nose rising under acceleration and dropping under braking). This is complicated stuff that is probably better mentioned only briefly but the normal reactions in the diagrams have been deliberately given different values in the different situations. Some possible scenarios are: 1 At rest or constant velocity 2 Accelerating with rear-wheel drive 3 Accelerating with front-wheel drive 4 Accelerating with 4-wheel drive 5 Decelerating under braking (or 4-wheel drive accelerating backwards) 6 Accelerating forwards with the rear handbrake on Draw the picture if the car has front-wheel drive and is accelerating backwards with the rear handbrake on. MEI Conference 2012 2 of 2 20/07/2012
Relative velocity In each case the train is moving at 0.2ms -1, relative to the table top. If the table is also moving with a constant velocity of 0.1ms -1, what is the resultant velocity of the train? In the last two cases, draw a velocity triangle and calculate the magnitude and direction of the resultant velocity. MEI Conference 2012 1 of 2 20/07/2012
Relative velocity Notes The relative velocity activity is designed to help students understand the wording of situations such as The velocity of a boat, relative to the water, is In this case, the train s velocity relative to the tray is 0.2ms -1, while the tray also moves with its own velocity. The resultant velocity can be found using a vector triangle, allowing students to practise this skill. As a visual aid, a toy train may be used with this activity. Ask students to estimate the velocity of the train (the one I used travelled approximately 20cm in 1 second, so I used 0.2ms -1 throughout the example). Put the train on a large tray (or table top), and when the train is moving in a straight line ask a series of questions such as What would the resultant velocity of the train be if I moved the tray at 0.1ms -1 in the same direction as the train? The idea of relative velocity can also be demonstrated dynamically (for instance using Geogebra), with a velocity triangle showing two vectors (in the case below, u and v) and a third vector (w) showing the relative velocity of v relative to u. MEI Conference 2012 2 of 2 20/07/2012
Drawing diagrams worksheet Draw diagrams for the following scenarios including all the forces acting with arrows and labels. Indicate any force whose direction may not be known. 1 A heavy box on a smooth slope. 2 A heavy box on a rough slope. 3 A heavy box on a rough slope being held in equilibrium by a string parallel to the slope. 4 A heavy box on a rough slope being pulled up the slope by a string parallel to the slope. 5 A heavy uniform ladder inclined at 20 to the vertical standing on a rough horizontal floor and resting against a smooth vertical wall. 6 As in 5 but with a rough vertical wall. C A B D 80 N The diagram shows a box of weight 80 N. BD is a light string in tension. The system is in equilibrium Draw diagrams for the following cases. 7 AC is a light string that passes through a small smooth ring attached to the box at B. 8 AB and BC are light strings, each tied to the box at B. MEI Conference 2012 1 of 1 20/07/2012
Drawing diagrams worksheet Draw diagrams for the following scenarios including all the forces acting with arrows and labels. Indicate any force whose direction may not be known. In the answers suggested: W is weight; R, R 1, S etc are normal reactions; T, T 1 etc are tensions; F, F 1 etc are frictional forces. 1 A heavy box on a smooth slope. R W 2 A heavy box on a rough slope. R F W 3 A heavy box on a rough slope being held in equilibrium by a string parallel to the slope. R T F W direction of F not known There is a lot to discuss in this example. The box may not be in limiting equilibrium and the direction of F depends on the relative magnitudes of T and of the component of W down the slope. 4 A heavy box on a rough slope being pulled up the slope by a string parallel to the slope. R T F W MEI Conference 2012 1 of 2 20/07/2012
5 A heavy uniform ladder inclined at 20 to the vertical standing on a rough horizontal floor and resting against a smooth vertical wall. 6 As in 5 but with a rough vertical wall. F 2 S S 20 20 W R W R F 1 F 1 Q 5 Q 6 The diagram shows a box of weight 80 N. BD is a light string in tension. The system is in equilibrium C A B D Draw diagrams for the following cases. 80 N 7 AC is a light string that passes through a small smooth ring attached to the box at B. 8 AB and BC are light strings, each tied to the box at B. A T N B T N C T 3 N D A T 1 N T 2 N B C T 4 N D 80 N 80 N Q 7 Q 8 MEI Conference 2012 2 of 2 20/07/2012
A kinematics example v m s -1 5 velocity 0 5 time t s 1 Draw the speed-time graph corresponding to the given velocity-time graph. With a suitable label for your graph, use the velocity axis for speed. 2 Find the areas between the velocity-time graph and the time axis. Interpret these as displacements. 3 The velocity-time graph describes the motion of an object in the interval 0 t 10. Initially ( i.e. when t = 0) the object is at A. The +ve direction is vertically upwards. For this object find (i) the direction of its motion when t = 2 and when t = 7, (ii) the distance it travels in the interval 0 t 10, (iii) its greatest displacement from A in the interval 0 t 10, (iv) the closest the object gets to A in the interval 4 t 10 and when this occurs. MEI Conference 2012 1 of 2 20/07/2012
1 velocity speed 0 time 2 Areas are 0.5 below, 6 above, 2.5 below, 0.5 above so displacements are 0 t 1 0.5 m vertically downwards 1 t 5 6 m vertically upwards 5 t 9 2.5 m vertically downwards 9 t 10 0.5 m vertically upwards 3 (i) At t = 2 +ve v so vertically upwards. At t = 7 -ve v so vertically downwards (ii) 0.5 6 2.5 0.5 9.5 m (iii) 0.5 + 6 = 5.5 m vertically upwards (after 5 seconds) (iv) At t = 4 it is 0.5 + 5.25 = 4.75 from A. Up to t = 5 it is getting further away ( by 0.75 m). From t = 5 to t = 9 it is getting closer ( by 2.5 m) and after t = 9 it is getting further away again. So closest is 4.75 + 0.75 2.5 = 3 m and this occurs when t = 9. MEI Conference 2012 2 of 2 20/07/2012
Sliding across the floor For an object sliding across a horizontal floor, investigate the relationship between the time it takes the object to come to rest and the distance it slides in that time. You should construct a mathematical model for the situation, devise an experiment to make direct measurements and then compare the experimental results with the prediction of the model. You have a stop watch, a measuring tape and an object. MEI Conference 2012 1 of 3 20/07/2012
Notes for teachers Assumptions: floor uniformly horizontal floor uniformly rough object doesn t bounce object doesn t spin no air resistance. General matters It is not possible to improve accuracy by repeating the experiment exactly. Could we scale this down and do it with coins on a table? The model R N direction of motion F N m kg mg N a m s -2 According to the assumptions, F is constant and there are no other forces. Applying N2L in the direction of motion F F = ma so a. m As F and M are constant, a is constant and we may use the constant acceleration formulae. MEI Conference 2012 2 of 3 20/07/2012
We would find it hard to find the initial speed of the object but do know it comes to rest so the suvat formula we need should involve s, v, a and t. 1 2 1 F 2 F 2 Using s vt at gives s 0 t t so s t 2 2 m 2m This means that a plot of s against t 2 should give a straight line through the origin. A refinement would be to find an expression for F in terms of the coefficient of friction using F R. Could we measure? Could this be a way of measuring? MEI Conference 2012 3 of 3 20/07/2012
Three forces in equilibrium Two pulleys are at the same height and in the same vertical plane. A string has objects P and Q with masses m 1 and m 2 attached to the two ends. It passes over both pulleys and a third object R, of mass m 3, is attached to the point X of the string between the pulleys. X With the system in equilibrium, the angles the two parts of the string attached to the central object make with the vertical are measured for various values of m 1, m 2 and m 3. Form a mathematical model for this situation and compare the predictions of the angles with the values measured. m 3 R m 1 P Q m 2 MEI Conference 2012 1 of 3 20/07/2012
Some questions that could be asked of students What assumptions are needed for a simple mathematical model and what are their relative importance? What measurements are necessary to be able to compare the predictions of the model with the experimental results? What problems do you think there would be in making sufficiently accurate measurements? Could you avoid the use of a protractor? Assumptions pulleys smooth strings light.. Need the string be inextensible? Difficulties with the measurements Some notes for teachers There is some difficulty in measuring the angles using a protractor students might find it easier to measure lengths and then use trig. Model T 1 X T 2 T 3 T 1 T 3 T 2 m 1 P R Q m 2 m 3 m 1 g m 3 g In the diagram take all the units to be SI m 2 g MEI Conference 2012 2 of 3 20/07/2012
Solution using the triangle of forces and the sine rule forces at X T 1 T 3 T 2 Using the sine rule T1 T2 T3 T3 sin sin a sin(180 ( )) sin( ) etc Of course, the problem may also be solved using resolution. Possible learning outcomes: Each different equilibrium position is associated with a particular position of the objects and there are limitations on the values of the masses. There are many other possibilities General matters How much help would your students need with creating the model? What the relative advantages and disadvantages of using a simulation instead of this experiment? MEI Conference 2012 3 of 3 20/07/2012
Why use simulations as well as or instead of experiments? What do we get from experiments? They allow us to consider the approximations to ideal behaviour by making assumptions such as lightness, smoothness, rigidity, inextensibility, the value of g and the suitability of Coulomb s law, Hooke s law etc. Students who have prior practical experience of a situation are more likely to recognise it and less likely to accept inappropriate answers Students have rich experience of mechanical situations e.g. cars (including learning to drive), lifts, boats, fun-fairs, watch F1 Experiments don t just provide new experiences they also help them interpret the ones they have already had. There can, however, be problems with using experiments to introduce students to standard situations. Experiments may be influenced by large but hidden factors, do not necessarily make it easy to explore a wide range of conditions, are sometimes very time consuming. What do we get from using simulations? Simulations allow students to familiarise themselves with the predictions of a mathematical model quickly over a wide range of controlled parameter changes interactively Projectile motion Experiments with projectiles to demonstrate non-resisted motion are very useful but suffer from the difficulty in reproducing starting conditions, the difficulty in seeing what is happening a projectile with slower motion usually has large air resistance MEI Conference 2012 1 of 2 20/07/2012
Some helpful simulations of projectiles moving with negligible air resistance 1. A projectile moving slowly on its trajectory with different angles and speeds of projection. 2. Complete trajectories and how they vary for different angles and speeds of projection. 3. The constancy of the horizontal component of velocity of a projectile and the changing nature of the vertical component. 4. The family of trajectories with the same initial speed but different angles of projection. This shows the envelope which is the parabola of safety and that there are two trajectories that pass through each point inside this envelope. 5. Two different trajectories with the same time of projection and the same initial speed. One can compare the progress of the projectile on the two trajectories. MEI Conference 2012 2 of 2 20/07/2012