Distance, Speed and Time

Transcription

1 Forces

2 Distance, Speed and Time D Speed = distance (in metres) time (in seconds) S T 1) Seb walks 200 metres in 40 seconds. What is his speed? 2) Lucy covers 2km in 1,000 seconds. What is her speed? 3) How long would it take Freddie to run 100 metres if he runs at 10m/s? 4) Sue travels at 50m/s for 20s. How far does he go? 5) Hannah drives her car at 85mph (about 40m/s). How long does it take her to drive 20km?

3 Distance-time graphs 2) Horizontal line = 40 4) Diagonal line downwards = Distance (metres) Time/s ) Diagonal line = 3) Steeper diagonal line =

4 40 Distance (metres) Time/s 1) What is the speed during the first 20 seconds? 2) How far is the object from the start after 60 seconds? 3) What is the speed during the last 40 seconds? 4) When was the object travelling the fastest?

5 Distance-time graph for non-uniform motion 40 Object is accelerating up to here Distance (metres) Object is now decelerating Time/s

6 Speed vs. Velocity Speed is simply how fast you are travelling This car is travelling at a speed of 20m/s Velocity is speed in a given direction (a vector quantity ) This car is travelling at a velocity of 20m/s east

7 4) A rocket accelerates from 1,000m/s to 5,000m/s in 2 seconds. What is its acceleration? Acceleration Acceleration = change in velocity (in m/s) (in m/s 2 ) time taken (in s) A V-U T 1) A cyclist accelerates from 0 to 10m/s in 5 seconds. What is her acceleration? 2) A ball is dropped and accelerates downwards at a rate of 10m/s 2 for 12 seconds. How much will the ball s velocity increase by? 3) A car accelerates from 10 to 20m/s with an acceleration of 2m/s 2. How long did this take?

8 Velocity-time graphs 1) Upwards line = 80 4) Downward line = Velocity m/s ) Horizontal line = 3) Upwards line = T/s

9 80 Velocity m/s T/s 1) How fast was the object going after 10 seconds? 2) What is the acceleration from 20 to 30 seconds? 3) What was the deceleration from 30 to 50s? 4) How far did the object travel altogether?

10 Speed-time graph for non-uniform motion 40 Object s acceleration is increasing Distance (metres) Object s acceleration is decreasing Time/s

11 Some subtle differences Distance is how far you have gone, displacement is how far you are and can be positive or negative: Distance = Displacement = Start Distance = Displacement = -1 metre 1 metre Distance = = Displacement = =

12 Some subtle differences Distance is how far you have gone, displacement is how far you are and can be positive or negative: Speed = Velocity = Speed = Velocity = Start -1 metre 1 metre Speed Speed = = Velocity = = Speed is how fast you go. Velocity is how fast in a given direction.

13 Vector vs. scalar Scalar quantities have size only and no direction. Vector quantities have both size and direction. Scalar or vector??? Scalar 8. Power Vector 2. Distance 12. Acceleration 1. Mass 6. Energy 7. Time 3. Displacement 4. Speed 11. Force 10. Current 5. Velocity 9. Momentum

14 Balanced and unbalanced forces Consider a camel standing on a road. What forces are acting on it? Reaction These two forces would be equal we say that they are BALANCED. The camel doesn t move anywhere. Weight

15 Balanced and unbalanced forces What would happen if we took the road away? Reaction The camel s weight is no longer balanced by anything, so the camel falls downwards Weight

16 Introduction to Forces A force is a push or a pull. Some common examples: Weight (mg) pulls things towards the centre of the Earth Air resistance/drag a contact force that acts against anything moving through air or liquid a contact force that acts against anything moving Upthrust keeps things afloat

17 Air Resistance Air resistance is a force that opposes motion through air. The quicker you travel, the bigger the air resistance: The same applies to a body falling through a liquid (called drag or upthrust ).

18 Examples of Air Resistance

19 Facts on Air Resistance Air Resistance is the force that acts anything that moves through the : It gets when you move faster: It can be reduced by using a more (aerodynamic) shape: Streamlining also works when travelling through : Words water, air, larger, against, streamlined

20 Balanced and unbalanced forces

21 Balanced and unbalanced forces 1) This animal is either or moving with 2) This animal is getting 3) This animal is getting. 4) This animal is

22 Resultant Force Calculate the resultant force of the following: 500N 100N 700N 600N 50N 700N 700N 800N 800N 200N 100N

23 Force and acceleration If the forces acting on an object are unbalanced then the object will accelerate, like these wrestlers: Force (in N) = Mass (in kg) x Acceleration (in m/s 2 ) F M A

24 Force, mass and acceleration 1) A force of 1000N is applied to push a mass of 500kg. How quickly does it accelerate? 2) A force of 3000N acts on a car to make it accelerate by 1.5m/s 2. How heavy is the car? 3) A car accelerates at a rate of 5m/s 2. If it weighs 500kg how much driving force is the engine applying? 4) A force of 10N is applied by a boy while lifting a 20kg mass. How much does it accelerate by? M F A

25 Terminal Velocity Consider a skydiver: 1) At the start of his jump the air resistance is so he downwards. 2) As his speed increases his air resistance will 3) Eventually the air resistance will be big enough to the skydiver s weight. At this point the forces are balanced so his speed becomes - this is called TERMINAL VELOCITY Words increase, small, constant, balance, accelerates

26 Terminal Velocity Consider a skydiver: 4) When he opens his parachute the air resistance suddenly, causing him to start. 5) Because he is slowing down his air resistance will again until it balances his. The skydiver has now reached a new, lower. Words slowing down, decrease, increases, terminal velocity, weight

27 Velocity-time graph for terminal velocity Velocity Speed increases Parachute opens diver slows down Terminal velocity reached On the Moon New, lower terminal velocity reached Time Diver hits the ground

28 A closer look at motion graphs Consider a bouncing ball: Displacement Time

29 A closer look at motion graphs Consider a bouncing ball: Velocity Time

30 A closer look at motion graphs Consider a bouncing ball: Acceleration Time

31 Equations of Motion If we re talking about any object travelling in a straight line with constant acceleration then we can use these 4 golden equations Golden equation #1 Golden equation #2 #1 Vel v Vel v Ave v-u u u T T Ave. velocity = (u + v) / 2 Acc = (v u) / t Therefore x = u + v 2 t Therefore v = u + at

32 Equations of Motion Golden equation #3 #1 Golden equation #4 #1 Vel v u v-u t(v-u)/2 ut T From equation #2 (v-u) = at Therefore x = ut + t/2 x at From eq n #2 t = (v-u) / a From eq n #1 x = t(u+v) / 2 So x = (v-u) (v+u) 2a 2ax = v 2 u 2 Therefore x = ut + ½at 2 Therefore v 2 = u 2 + 2ax

33 Equations of Motion x = u + v 2 t v = u + at x = ut + ½at 2 They re golden! v 2 = u 2 + 2ax

34 Example questions 1) Ben drops a ball on Dan s foot. How long does the ball take to fall 1m? 2m? Why is the second answer not twice the first? 2) Ryan flies to Belgium. His aeroplane has a maximum acceleration on the ground of 3.4ms -2. What is the minimum length of runway needed to reach its take off speed of 110ms -1 and how long will this take? 3) Luke likes watching kangaroos. A kangaroo jumps to a vertical height of 2.8m. For how long was it in the air? 4) Tom likes baseball. A baseball pitcher can release a ball at 40ms -1 after accelerating through a distance of 2.5m. Calculate the average acceleration of the ball.

35 7) Harry has a good chance of surviving a car crash with a seatbelt on if his deceleration does not exceed 30g. Calculate the distance by which the front end of the car must collapse in if a crash occurs at 70mph. Example questions 5) Andrew wants to play with the air track. The air track is slightly tilted. He pushes a trolley up the track with a speed of 1ms -1 and its acceleration due to the tilt is 0.5ms -2 down the track. How long does it take to drop 1m below the starting point? 6) Howard travels in a rocket powered sledge and accelerates from rest to 284ms -1 in 5s and then comes to a rest in 1.5s. Calculate his acceleration in both stages.

36 x = 81.5m Vertical Projection If I throw this ball upwards with a speed of 40ms -1 how high will it go? Use v 2 = u 2 + 2ax 0 = (2 x x x) 0 = x 1600 = 19.62x x = 1600/19.62

37 4) A test tube falls off the table. If the table is 1m high how fast was the test tube going when it hit the floor? Practice Questions 1) How far will a cricket ball go if it is thrown upwards with an initial velocity of 10ms -1? 2) How far will a table tennis ball go if it is thrown upwards with an initial velocity of 5ms -1? 3) A human cannonball is projected vertically upwards and she reaches a vertical height of 20m before coming back down. How fast was she going when she left the cannon?

38 Measuring g Consider the equation x = ut + ½at 2 If we consider a ball being dropped then u=0, so x = ½at 2 We also know that a = g, therefore x = ½gt 2 x x x = ½ g t 2 x y = m x + c x x Gradient = g ½t 2

39 Projectile Motion Aha! If I let go of the branch when he fires his gun I ll be safe because the bullet will go above me

40 Projectile Motion Question how long did this take and how fast was the bullet? 1.5m 50m 1) Use x = ut + ½at 2 vertically to find the time 2) Then use speed = distance / time horizontally to get the

41 Example questions 1) Ben throws a bowling ball at Tom and it lands on his foot. If the ball started 1.2m above Tom s foot and the distance between them was 2m calculate both the time taken and the initial velocity of the ball. 2) Rob fires a gun and the bullet leaves the barrel at a speed of 200ms -1. If it landed on the ground 500m away calculate how long the journey took and how high up Rob was holding the gun from ground level. 3) Andrew likes knocking test tubes off the table. If he hits one with an initial velocity of 2ms -1 and the table is 1m high calculate the time taken for the test tube to hit the floor

42 Recap questions 1) Andrew Murray hits a tennis ball and it passes horizontally over the net and lands just inside the baseline of the court. The net has a height of 1.07m and is 11.9m from the baseline. Find the horizontal speed of the ball. 2) David Beckham takes a free kick and it flies into the top corner horizontally. If the corner is 2.4m above the ground and the goal is 18m away calculate the time taken for the ball to reach the goal.

43 Newton s 1 st Law of Motion Basically, a body will remain at rest or continue to move with constant velocity as long as the forces acting on it are balanced. and an unbalanced backwards force will make me slow down An unbalanced forwards force will make me accelerate Newton

44 Newton s 2 nd Law of Motion The acceleration of a body is proportional to the resultant force causing its acceleration and is in the same direction. Newton In other words force = mass x acceleration F M A

45 Revision questions 1) A force of 1000N is applied to push a mass of 500kg. How quickly does it accelerate? 2) A force of 3000N acts on a car to make it accelerate by 1.5ms -2. How heavy is the car? 3) A car accelerates at a rate of 5ms -2. If it weighs 500kg how much driving force is the engine applying? 4) A force of 10N is applied by a boy while lifting a 20kg mass. How much does it accelerate by? M F A

46 Testing Newton s 2 nd Law For each version of the experiment: 1) Draw a diagram of how you set it up 2) Describe your method 3) Describe what equipment you used to get the results and how you analysed them (you only need to do this once as they re both the same).

47 Newton s 3 rd Law of Motion When body A exerts a force on body B, body B exerts an equal and opposite force on body A. My third law says that if I push to the right I will move backwards as well. Newton

48 Newton s 3 rd Law of Motion What will happen if I push this satellite away from me?

49 Types of Forces Gravitational (W=mg) Electromagnetic/ electrostatic Nuclear (2 types) Describe each force, including a comment on the distance it works over, whether it repels or attracts and other important points

50 Free body force diagrams The Earth pulls Newton down with a gravitational force of 700N. what on what type direction size Newton pulls the Earth up with a gravitational force of 700N. This is a Newton III pair of forces

51 Free body force diagrams 2 Consider a man standing on a table on the Earth:

52 Newton I vs. Newton III These two forces are acting on the same body, they re two different types of force and the man is in equilibrium as long as the forces balance this is a Newton I pair of forces. These two forces are acting on different bodies, they re both the same type and they are always equal and opposite this is a Newton III pair of forces.

53 Summary Newton I A law about the forces on Concerns any of forces The forces can be types If there are two forces and the body is in equilibrium the forces are and Newton I only applies when the body is in Newton III A law about the forces on Always concerns forces only Both forces are type The two forces are ALWAYS and Newton III applies

54 Random recap questions 1) Nick runs the last 100m of a 200m race over 15s. If he was accelerating at a rate of 1ms -2 during those 15s how fast was he running when he passed the 100m mark? 2) Ben throws a ball from the 1 st floor at Ryan below. If the ball travels for 1.5s before hitting Ryan how far above Ryan is Ben? If Ryan is 20m away from the building how fast did Ben throw it? 3) Dan is swinging a conker around on a piece of string. Draw a free body force diagram for each object (you may find it easier to draw both on the same diagram). 4) For each of the forces in the previous two diagrams identify the Newton III pair and describe what the force is, what is acts on and its direction.

55 Vectors 10km 10km 14.1km 100ms -1 5ms ms -1

56 F 2 = F cos θ Resolving Vectors Consider a diagonal push: This force is given by: F F 1 = F sin θ θ F 1 F 2 This force is given by:

57 Resolving Vectors example questions Calculate the horizontal and vertical components of the following: 1) 10N 2) 20N 50 O 35 O Work out the size and direction of the resultant force: 3) 4) 20N 8N 10N 15N 50 O 45 O 80 O 30 O

58 Free body force diagrams 3 Consider a man on a sloping table: Reaction (a contact force) is perpendicular to the surface. Friction (a tangential contact force) goes up the slope. Let s combine the forces Resultant force is zero, so no acceleration

59 Free body force diagrams 1) Draw a free body force diagram for a ladder against a wall. 2) A car pulls a caravan along the M25. Draw a free body force diagram for the caravan. 3) Draw a free body force diagram for a 4-wheel drive car driving up the M1. 4) Draw a free body force diagram for a 2-wheel drive (engine at the front) car driving up the M1 as well.

60 Stopping a car Tiredness Too many drugs Thinking distance (reaction time) Too much alcohol Poor visibility Icy roads Tyres/brakes worn out Braking distance Wet roads Driving too fast

61 4) On the moon the gravitational field strength is 1.6N/kg. What will Dave weigh if he stands on the moon? Weight vs. Mass Earth s Gravitational Field Strength is 10N/kg. In other words, a 1kg mass is pulled downwards by a force of 10N. Weight = Mass x Gravitational Field Strength W (in N) (in kg) (in N/kg) M g 1) What is the weight on Earth of a book with mass 2kg? 2) What is the weight on Earth of an apple with mass 100g? 3) Dave weighs 700N. What is his mass?

62 Work done When any object is moved around work will need to be done on it to get it to move (obviously). We can work out the amount of work done in moving an object using the formula: Work done = Force x distance moved in J in N in m W F D

63 Example questions 1. Bori pushes a book 5m along the table with a force of 5N. He gets tired and decides to call it a day. How much work did he do? 2. Alicia lifts a laptop 2m into the air with a force of 10N. How much work does she do? 3. Martin does 200J of work by pushing a wheelbarrow with a force of 50N. How far did he push it? 4. Chris cuddles his cat and lifts it 1.5m in the air. If he did 75J of work how much force did he use? 5. Carl drives his car 1000m. If the engine was producing a driving force of 2000N how much work did the car do?

64 Elastic Potential Energy Elastic potential energy is the energy stored in a system when work is done to change its shape, e.g:

65 Kinetic energy Any object that moves will have kinetic energy. The amount of kinetic energy an object has can be found using the formula: Kinetic energy = ½ x mass x velocity squared in J in kg in m/s KE = ½ mv 2

66 Example questions 1) Nicole drives her car at a speed of 30m/s. If the combined mass of her and the car is 1000kg what is her kinetic energy? 2) Shanie rides her bike at a speed of 10m/s. If the combined mass of Shanie and her bike is 80kg what is her kinetic energy? 3) Dan is running and has a kinetic energy of 750J. If his mass is 60kg how fast is he running? 4) George is walking to town. If he has a kinetic energy of 150J and he s walking at a pace of 2m/s what is his mass?

67 Random questions 1) Sophie tries to run 100m in 12 seconds and succeeds. How fast did she run? 2) Tommy accelerates at a rate of 2m/s 2 for 3 seconds. If he started at 10m/s what was his final speed? 3) Charlie decides to lift his book up into the air. His book has a mass of 100g and he lifts it 50cm. Calculate the work done. 4) Lewis accelerates from 0 to 10m/s in 5 seconds. If his mass is 70kg how much force did his legs apply? 5) Rachel rides 1km at a speed of 20m/s. How long did the journey take? 6) Claire thinks it s funny to push James with a force of 120N. If James has a mass of 60kg calculate his acceleration. 7) Lauren slams on the brakes on her bike and her brakes do 20,000J of work. If the combined mass is 100kg what speed was she travelling at? 8) Tom has a mass of 75kg. If he accelerates from 10 to 20m/s in 2s how much force did he apply?

68 Random questions 9) Georgina amuses herself by throwing things at Sarah. If she throws a ball with a speed of 20m/s and the distance between her and Sarah is 5m how long will it take to reach her? 10) Mr Richards throws calculators around the room with a force of 20N. If each calculator has a mass of 200g calculate the acceleration. 11) Sam has a mass of 70kg. What is his weight on Earth, where the gravitational field strength is 10N/kg? 12) Zak does some work by pushing a box around with a force of 1N. He does 5J of work and decides to call it a day. How far did he push it? 13) On the moon Matt might weigh 112N. If the gravitational field strength on the moon is 1.6N/kg what is his mass? What will he weigh on Earth? 14) Dan likes bird watching. He sees a bird fly 100m in 20s. How fast was it flying? 15) How much kinetic energy would Richard have if he travelled at a speed of 5m/s and has a mass of 70kg?

69 Momentum Any object that has both mass and velocity has MOMENTUM. Momentum (symbol p ) is simply given by the formula: P Momentum = Mass x Velocity (in kgms -1 ) (in kg) (in ms -1 ) M V What is the momentum of the following? 1) A 1kg football travelling at 10ms -1 2) A 1000kg Ford Capri travelling at 30ms -1 3) A 20g pen being thrown across the room at 5ms -1 4) A 70kg bungi-jumper falling at 40ms -1

70 Conservation of Momentum In any collision or explosion momentum is conserved (provided that there are no external forces have an effect). Example question: Two cars are racing around the M25. Car A collides with the back of car B and the cars stick together. What speed do they move at after the collision? Speed = 50ms -1 Speed = 20ms -1 Mass = 1000kg Mass = 800kg Mass = 1800kg Speed =??ms -1 Momentum before = momentum after so 1000 x x 20 = 1800 x V V = 36.7ms -1

71 Momentum in different directions What happens if the bodies are moving in opposite directions? Speed = 50ms -1 Speed = 20ms -1 Mass = 1000kg Mass = 800kg Momentum is a VECTOR quantity, so the momentum of the second car is negative Total momentum = 1000 x x 20 = kgms -1 Speed after collision = kgms -1 / 1800 = 18.9ms -1

72 Another example Consider the nuclear decay of Americium-241: 237 Np Am 4 2 α If the new neptunium atom moves away at a speed of 5x10 5 ms -1 what was the speed of the alpha particle?

73 More questions 1) A white snooker ball moving at 5m/s strikes a red ball and pots it. Both balls have a mass of 1kg. If the white ball continued in the same direction at 2m/s what was the velocity of the red ball? 2) A car of mass 1000kg heading up the M1 at 50m/s collides with a stationary truck of mass 8000kg and sticks to it. What velocity does the wreckage move forward at? 3) A defender running away from a goalkeeper at 5m/s is hit in the back of his head by the goal kick. The ball stops dead and the player s speed increases to 5.5m/s. If the ball had a mass of 500g and the player had a mass of 70kg how fast was the ball moving? 4) A gun has a recoil speed of 2m/s when firing. If the gun has a mass of 2kg and the bullet has a mass of 10g what speed does the bullet come

74 2) The speed the ball moves away with Force and momentum Newton s second law of motion says that the force acting on an object is that object s rate of change of momentum. In other words Force = Change in momentum (in N) Time (in s) Also called impulse (in kgm/s) F mv T For example, David Beckham takes a free kick by kicking a stationary football with a force of 40N. If the ball has a mass of 0.5kg and his foot is in contact with the ball for 0.1s calculate: 1) The change in momentum of the ball (its impulse),

75 Example questions 1) Ben likes playing golf. He strikes a golf ball with a force of 80N. If the ball has a mass of 200g and the club is in contact with it for 0.2s calculate a) the change in momentum of the golf ball, b) its speed. 2) Nick thinks it s funny to hit tennis balls at Tom. He strikes a serve with a force of 30N. If the ball has a mass of 250g and the racket is in contact with it for 0.15s calculate the ball s change in momentum and its speed. 3) Dan takes a dropkick by kicking a 0.4kg rugby ball away at 10m/s. If his foot was in contact with the ball for 0.1 seconds calculate the force he applied to the ball. 4) Simon strikes a 200g golf ball away at 50m/s. If he applied a force of 50N calculate how long his club was in contact with the ball for.

76 Safety features Let s use Newton s Second Law to explain how airbags work: mv F T Basically: 1) The change in momentum is the same with or without an airbag 2) But having an airbag increases the time of the collision 3) Therefore the force is reduced

77 Car Safety Features

78 Energy loss in collisions In the Forces module we looked at how to calculate an object s kinetic energy: Kinetic energy = ½ x mass x velocity squared in J in kg in m/s We ve also said that in a collision momentum is conserved (unless an external force acts). The same cannot usually be said for kinetic energy For example, consider the following collision. How much kinetic energy is lost? Before Speed = 50m/s Speed = 20m/s Mass = 1000kg Mass = 800kg After Mass = 1000kg Speed = 20m/s Mass = 800kg Speed = 30m/s

79 Energy loss in collisions Consider a head-on collision where the cars stick together. How much kinetic energy is lost in this example? Where does all the energy go? Before Speed = 50m/s Speed = 30m/s m=800kg After m=3000kg Speed = 10m/s In this example more kinetic energy was lost. We say it was a less elastic collision. An elastic collision is one where the kinetic energy is conserved.

80 Conservation of Energy Consider a bouncing ball: Gravitational Potential Energy Time

81 Conservation of Energy Consider a bouncing ball: Kinetic Energy Time

82 Conservation of Energy Now put these graphs together: Kinetic Energy Total energy of the ball Time

83 Finding the Centre of Mass Definition: The Centre of mass of an object is the point at which all the mass of the object is centred. Q. How would you find the centre of mass of this object?

84 Stability 1. Centre of mass is within the wheelbase no problem! 2. Centre of mass is directly above the edge of the wheelbase car is on the point of toppling 3. Car falls over

85 Turning Moments A moment is a turning force, e.g. trying to open or close a door or using a spanner. The size of the moment is given by: Moment (in Nm) = force (in N) x PERPENDICULAR distance from pivot (in m) Calculate the following turning moments: 5 metres 100 Newtons 2 metres 200 Newtons

86 Turning Moments 2 metres 2 metres 200 Newtons 100 Newtons Total ANTI-CLOCKWISE turning moment = 200x2 = 400Nm Total CLOCKWISE turning moment = 100x2 = 200Nm The anti-clockwise moment is bigger so the seesaw will turn anti-clockwise

87 An example question 5 metres? metres 2000 Newtons 800 Newtons

88 Calculate the missing quantity The following are all balanced: 2N??N 4m 2m 5N 3N 2m??m 5N 5N 15N 4m??m 2m

89 Balanced or unbalanced?

90 A hard question Consider a man walking along a plank of wood on a cliff. ow far can he walk over the cliff before the plank tips over? Aaarrgghh Man s weight = 800N 3m 1m Plank s weight = 200N

91 A recap question Calculate the mass of man in the example given below: 30kg 0.4m 1.2m

92 Centripetal force Consider a ball of Pleistocene attached to some string: The ball is kept in its path by the tension in the string an example of a CENTRIPETAL FORCE. This force also produces the change in velocity due to the direction constantly changing. This force is INCREASED if you increase the mass of the object, its speed or decrease the radius of the circle. Other examples of centripetal forces: Orbits Electrons

93 Centripetal force and The Earth Gravity (and the fact that the Earth is moving at high speeds) keeps the Earth in orbit. Notice that the orbit path is slightly elliptical

94 Different Orbits Mercury Venus Mercury = 88 days Mars = 687 days Earth Mars Jupiter Saturn Uranus Pluto = 90,500 days Neptune Pluto

95 Gravity Gravity is an attractive force that affects anything with mass: Note that this force goes both ways the Earth is attracted to us.

96 Gravity on different planets: Earth gravitational field strength = 9.8N/Kg Jupiter gravitational field strength = 25N/Kg

97 More information on gravity The amount of gravity attracting an object decreases the further out the object is F If you double the distance the gravitational force divides by 4 F/4 F/9 If you triple the distance the force divides by 9

98 Density Density = Mass Volume ρ = m V 1) What is the density of a piece of wood of volume 2m 3 and mass 1200kg? 2) Air only has a density of 1.3kg/m 3. What is the mass of 0.2m 3 of air? 3) Carbon dioxide is more dense and the same volume would have a mass of 0.38kg. What is its density? 4) The mercury in a thermometer has a volume of 5x10-5 m 3. If the density of the mercury is 13600kg/m 3 what mass of mercury is in the thermometer?

99 The metal block will because it is dense than water The wooden block will because it is dense than water Floating Whether or not an object will float depends on its DENSITY. For example:

100 Pressure some basic ideas Describe and draw experiments for each of the following ideas: 1) Pressure increases with depth 2) Pressure is the same in all directions at a certain depth

101 Pressure Explain how the following work: 1) A sharp knife is easier to use then a blunt knife 2) A drawing pin 3) A woman with stilettos on might damage a floor more than an elephant would

102 Pressure Pressure depends on two things: 1) How much force is applied, and 2) How big (or small) the area on which this force is applied is. Pressure can be calculated using the equation: Pressure (in N/m 2 ) = Force (in N) F OR in cm 2 and N/cm 2 Area (in m 2 ) P A

103 Some pressure questions 1) Calculate the pressure exerted by a 1000N elephant when standing on the floor if his feet have a total area of 1m 2. 2) A brick is rested on a surface. The brick has an area of 20cm 2. Its weight is 10N. Calculate the pressure. 3) A woman exerts a pressure of 100N/cm 2 when standing on the floor. If her weight is 500N what is the area of the floor she is standing on? 4) (Hard!) The pressure due to the atmosphere is 100,000N/m2. If 10 Newtons are equivalent to 1kg how much mass is pressing down on every square centimetre of our body?

104 Some examples of pressure

105 Gas Pressure

106 Hydraulic systems

107 Levers Pivot Effort Load

108 3 types of lever Load is on other side of pivot (e.g. seesaw) Load is outside the effort, (e.g. biceps) Load is inside the effort (e.g. wheelbarrow)

109 Circular Motion

110 Circular Motion 1) Is this car travelling at constant speed? 2) Is this car travelling at constant velocity?

111 Centripetal Acceleration If the velocity is changing then it must be accelerating... V a ΔV This change in velocity is towards the centre of the circle so the acceleration and is towards the centre if the circle Centripetal Acceleration V b

112 Radians To further understand circular motion we need to use a different system for measuring angles: Old method New method r s Angle = 30 0 Angle = s/r rad

113 Radians Calculate the following angles in radians: 1) 1.5cm 2) 15cm 2cm 6cm 3) 4) 2.05cm 50.24m 5mm 8m

114 Centripetal Acceleration Consider a circle: r v 1 v 2 θ θ v 1 v 2 Δv If we assume θ is very small then v 1 = v 2 = v Therefore θ = Δv/v Also θ = vδt/r Therefore Δv/v = vδt/r Δv/Δt = v 2 /r a = v 2 /r

115 More Exciting Equations From the last slide a = v 2 /r but F=ma so centripetal force F = mv 2 /r The angular speed is the angular distance divided by time, or ω = θ/t The total time period T for one revolution must therefore be the time taken to complete 2π revolutions, or ω = 2π/T Frequency is how often something happens every second, so T = 1/f. Therefore ω = 2πf For a whole circle, v = 2πr/T. However, T = 2π/ω. Therefore v = rω Acceleration a = v 2 /r, therefore a = rω 2 Finally, this must mean that F = mrω 2 F = mv 2 /r ω = θ/t ω = 2π/T ω = 2πf v = rω a = rω 2 F = mrω 2

116 6) If the combined mass of Tom and his car is 1000kg calculate the centripetal force if his turning circle has a radius of 3m. Example questions 1) A disc spins twice per second. Calculate its angular speed. 2) Estimate the angular speed of the Earth. 3) Scoon spins a conker around his head using a 50cm long string. The conker has a mass of 0.1kg and he spins it with a velocity of 2ms -1. Calculate the centripetal force. 4) Calculate the velocity of a satellite moving with an angular speed of 7x10-5 rads -1 and at an altitude of 700km above the Earth (radius 6370km). 5) Tom drives his car in circles. If he drives with an angular speed of 1 rads -1 how many times will he make a complete turn in 10 seconds?

117 Simple Harmonic Motion Definition: simple harmonic motion is when acceleration is proportional to displacement and is always directed towards equilibrium.

118 Simple Harmonic Motion Consider a pendulum bob: Let s draw a graph of displacement against time: Equilibrium position Displacement Sinusoidal Time

119 Displacement SHM Graphs Time Velocity Time Acceleration Time

120 The Maths of SHM Displacement As we ve already seen, SHM graphs are sinusoidal in shape: Time Therefore we can describe the motion mathematically as: x = x 0 cosωt v = -x 0 ωsinωt a = -ω 2 x a = -x 0 ω 2 cosωt

121 The Maths of SHM Recall our definition of SHM: Definition: simple harmonic motion is when acceleration is proportional to displacement and is always directed towards equilibrium. This agrees entirely with the maths: a a = -ω 2 x x Important remember ω = 2π/T

122 2 a 5 SHM questions x 1) Calculate the gradient of this graph 2) Use it to work out the value of ω 3) Use this to work out the time period for the oscillations 4) Howard sets up a pendulum and lets it swing 10 times. He records a time of 20 seconds for the 10 oscillations. Calculate the period and the angular speed ω. 5) The maximum displacement of the pendulum is 3cm. Sketch a graph of a against x and indicate the maximum acceleration. a x

123 a max = -ω 2 x 0 SHM Maximum Values Consider our three SHM equations: x = x 0 cosωt v = -x 0 ωsinωt a = -x 0 ω 2 cosωt Clearly, the maximum value that sinωt can take is 1, therefore: x max = x 0 (obviously) v max = -x 0 ω (or max speed = ωx 0 )

124 SHM periods: Two examples For a pendulum the only thing that affects the period is the length of the string: T = 2π l g

125 SHM periods: Two examples For a spring there are two things that affect the period the mass and the spring constant: T = 2π m k Where k is defined as the force needed to extend the spring by a given number of metres (units Nm -1 ): F = -kδx

126 More questions 1) Define simple harmonic motion. 2) A pendulum in a grandfather clock has a period of 1 second. How long is the pendulum? 3) Luke sets up a 200g mass on a spring and extends it beyond its equilibrium. He then releases it and enjoys watching it bounce up and down. If the period is 10s what is the spring constant? 4) Nick is envious of this and sets up another system with a spring constant of 0.1Nm -1. If the spring oscillates every 8 seconds how much mass did he use? 5) Simon sets up a pendulum and records the period as being 3 seconds. He then lengthens the pendulum by 1m and does the experiment again. What is the new period?

127 SHM recap questions 1) Define SHM and state the golden SHM equation 2) A body is performing SHM and is temporarily at rest at time t=0. Sketch graphs of its displacement, velocity and acceleration. 3) A body is performing SHM as shown on this graph. Calculate its angular speed and its time period T. 4) What is this body s maximum speed? 5 a 10 x 5) A 1kg mass is attached to a spring of spring constant 10Nm -1. The mass is pulled down by 5cm and released. It performs SHM. Calculate the time period of this motion. 6) Describe the energy changes in this system as it bounces up and down. 7) Calculate the length of a pendulum if it oscillates with a period of 5s.

128 SHM: Energy change Equilibrium position Energy GPE K.E. Time