The law of Conservation of Energy. Monday, October 31, 11

The law of Conservation of Energy

Force as the derivative of potential energy We often encounter situations where we know the potential energy as a function of position and we need to find the corresponding force. We recall that for a conservative force: W = ΔU So if we now apply this to a small displacement Δx, the work done by the force Fx(x) during this displacement approaches Fx(x) Δx as Δx approaches zero. F x (x)δx = ΔU F x (x) = ΔU Δx So in the limit as Δx 0 F x (x) = du(x) dx

Force as the derivative of potential energy F x (x)δx = ΔU F x (x) = ΔU Δx So in the limit as Δx 0 F x (x) = du(x) dx

Energy diagrams give us insight Energy diagrams plot energy as a function of position. These diagrams give useful information about limits and zeros for the physical properties involved.

The potential energy curve for motion of a particle F x = du dx

Energy diagrams give us insight

Summary

Q7.1 A piece of fruit falls straight down. As it falls, A. the gravitational force does positive work on it and the gravitational potential energy increases. B. the gravitational force does positive work on it and the gravitational potential energy decreases. C. the gravitational force does negative work on it and the gravitational potential energy increases. D. the gravitational force does negative work on it and the gravitational potential energy decreases.

A7.1 A piece of fruit falls straight down. As it falls, A. the gravitational force does positive work on it and the gravitational potential energy increases. B. the gravitational force does positive work on it and the gravitational potential energy decreases. C. the gravitational force does negative work on it and the gravitational potential energy increases. D. the gravitational force does negative work on it and the gravitational potential energy decreases. W = ΔU

Q7.2 You toss a 0.150-kg baseball straight upward so that it leaves your hand moving at 20.0 m/s. The ball reaches a maximum height y 2. What is the speed of the ball when it is at a height of y 2 /2? Ignore air resistance. A. 10.0 m/s v 2 = 0 v 1 = 20.0 m/s m = 0.150 kg y 2 y 1 = 0 B. less than 10.0 m/s but more than zero C. more than 10.0 m/s D. not enough information given to decide

A7.2 You toss a 0.150-kg baseball straight upward so that it leaves your hand moving at 20.0 m/s. The ball reaches a maximum height y 2. What is the speed of the ball when it is at a height of y 2 /2? Ignore air resistance. A. 10.0 m/s v 2 = 0 v 1 = 20.0 m/s m = 0.150 kg y 2 y 1 = 0 B. less than 10.0 m/s but more than zero C. more than 10.0 m/s D. not enough information given to decide

Q7.4 The two ramps shown are both frictionless. The heights y 1 and y 2 are the same for each ramp. A block of mass m is released from rest at the left-hand end of each ramp. Which block arrives at the righthand end with the greater speed? A. the block on the curved track B. the block on the straight track C. Both blocks arrive at the right-hand end with the same speed. D. The answer depends on the shape of the curved track.

A7.4 The two ramps shown are both frictionless. The heights y 1 and y 2 are the same for each ramp. A block of mass m is released from rest at the left-hand end of each ramp. Which block arrives at the righthand end with the greater speed? A. the block on the curved track B. the block on the straight track C. Both blocks arrive at the right-hand end with the same speed. D. The answer depends on the shape of the curved track.

Q7.5 A block is released from rest on a frictionless incline as shown. When the moving block is in contact with the spring and compressing it, what is happening to the gravitational potential energy U grav and the elastic potential energy U el? A. U grav and U el are both increasing. B. U grav and U el are both decreasing. C. U grav is increasing, U el is decreasing. D. U grav is decreasing, U el is increasing. E. The answer depends on how the block s speed is changing.

A7.5 A block is released from rest on a frictionless incline as shown. When the moving block is in contact with the spring and compressing it, what is happening to the gravitational potential energy U grav and the elastic potential energy U el? A. U grav and U el are both increasing. B. U grav and U el are both decreasing. C. U grav is increasing, U el is decreasing. D. U grav is decreasing, U el is increasing. E. The answer depends on how the block s speed is changing.

Q7.6 The graph shows the potential energy U for a particle that moves along the x-axis. The particle is initially at x = d and moves in the negative x- direction. At which of the labeled x-coordinates does the particle have the greatest speed? A. at x = a B. at x = b C. at x = c D. at x = d E. more than one of the above

A7.6 Potential energy lowest, The graph shows the potential kinetic energy highest energy U for a particle that moves along the x-axis. The particle is initially at x = d and moves in the negative x- direction. At which of the labeled x-coordinates does the particle have the greatest speed? A. at x = a B. at x = b C. at x = c D. at x = d E. more than one of the above

Q7.7 The graph shows the potential energy U for a particle that moves along the x-axis. The particle is initially at x = d and moves in the negative x- direction. At which of the labeled x-coordinates is the particle slowing down? A. at x = a B. at x = b C. at x = c D. at x = d E. more than one of the above

A7.7 The graph shows the potential energy U for a particle that moves along the x-axis. The particle is initially at x = d and moves in the negative x- direction. At which of the labeled x-coordinates is the particle slowing down? The slope is positive so the acceleration is negative F x = du dx = ma x A. at x = a B. at x = b C. at x = c D. at x = d E. more than one of the above

Q7.8 The graph shows the potential energy U for a particle that moves along the x-axis. At which of the labeled x-coordinates is there zero force on the particle? A. at x = a and x = c B. at x = b only C. at x = d only D. at x = b and d E. misleading question there is a force at all values of x.

A7.8 The graph shows the potential energy U for a particle that moves along the x-axis. At which of the labeled x-coordinates is there zero force on the particle? The slope is zero so the force is zero F x = du dx A. at x = a and x = c B. at x = b only C. at x = d only D. at x = b and d E. misleading question there is a force at all values of x.

Q7.9 The graph shows a conservative force F x as a function of x in the vicinity of x = a. As the graph shows, F x = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? 0 F x a x A. U = 0 at x = a B. U is a maximum at x = a. C. U is a minimum at x = a D. U is neither a minimum or a maximum at x = a, and its value at x = a need not be zero.

A7.9 The graph shows a conservative force F x as a function of x in the vicinity of x = a. As the graph shows, F x = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? 0 F x a x A. U = 0 at x = a B. U is a maximum at x = a. C. U is a minimum at x = a. D. U is neither a minimum or a maximum at x = a, and its value at x = a need not be zero.

Q7.10 The graph shows a conservative force F x as a function of x in the vicinity of x = a. As the graph shows, F x = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? 0 F x a x A. U = 0 at x = a B. U is a maximum at x = a. C. U is a minimum at x = a. D. U is neither a minimum or a maximum at x = a, and its value at x = a need not be zero.

A7.10 The graph shows a conservative force F x as a function of x in the vicinity of x = a. As the graph shows, F x = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? 0 F x a x A. U = 0 at x = a B. U is a maximum at x = a. C. U is a minimum at x = a. D. U is neither a minimum or a maximum at x = a, and its value at x = a need not be zero.

Q7.11 The graph shows a conservative force F x as a function of x in the vicinity of x = a. As the graph shows, F x > 0 and df x /dx < 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? 0 F x a x A. du/dx > 0 at x = a B. du/dx < 0 at x = a C. du/dx = 0 at x = a D. Any of the above could be correct.

A7.11 The graph shows a conservative force F x as a function of x in the vicinity of x = a. As the graph shows, F x > 0 and df x /dx < 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? 0 F x a x A. du/dx > 0 at x = a B. du/dx < 0 at x = a C. du/dx = 0 at x = a D. Any of the above could be correct. F x = du dx At a Fx>0 so du/dx<0

Momentum, Impulse and Angular Momentum

Introduction If you watch a football game, you ll see collisions, tackles, many men colliding at once, maybe just two in an open area. Are these situations different? Newton told us the forces result in acceleration of a mass. We will now study two new points of view momentum and impulse. F = ma = m d v dt = d dt (m v) = d p dt

How does momentum relate to mass and velocity? Understanding momentum begins with the simple relationship that momentum is equal to mass multiplied by velocity.

Impulse F = ma = d p dt = p p 2 1 t 2 t 1 F(t t ) 2 1 = p 2 p 1 = J

Impulse

The meaning of the area of a F versus t graph The impulse is the area under the F versus t graph

Impulse Although impulse and momentum are both vector quantities, it is often easier to deal with the components.

Compare momentum and kinetic energy The work-energy theorem tells us that a change in a particle s kinetic energy is due to work done on the particle W tot = K 2 K 1 The impulse-momentum theorem tells us that a change in a particle s momentum is due to an impulse J = p 2 p 1

Compare momentum and kinetic energy The work-energy theorem focuses on the distance of force application. The impulse momentum relationship depends on duration of an impact. Both rest on the foundation of Newton s laws and are integral principles, relating the motion at two different times separated by a finite interval. Newton s second law is a differential principle, relating forces to the rate of change of either velocity or momentum.

A ball hits a wall

A ball hits a wall (a) J = p p 2 1

A ball hits a wall

A ball hits a wall (b) F(t t ) 2 1 = p 2 p 1 = J

F av = J Δt Duration of an impact F(t t ) 2 1 = p 2 p 1 = J

F av = J Δt Duration of an impact The impulse momentum relationship depends on the duration of an impact. Golf ball hitting a steel plate at 150 mph shot at 70,000 fps F(t t ) 2 1 = p 2 p 1 = J

Kicking a soccer ball

Like energy, momentum also has conservation rules

Conservation of Momentum means Conservation of its components If the vector sum of the external forces on the system is zero, then the components of momentum are all constant.

Recoil of a rifle

Recoil of a rifle Conservation of momentum in the x-direction gives:

Recoil of a rifle

Recoil of a rifle W tot = K 2 K 1 The Kinetic Energy gained is: F s, where s is the displacement. During the period of interaction between the bullet and the rifle, t, the bullet travelled much farther than the rifle, so the bullet acquires much greater kinetic energy than the rifle. Work is F s, so the bullet does more work than the rifle. The ratio of the two kinetic energies is 600:1, this is equal to the inverse ratio of their masses (get this from conservation of momentum). This always happens, if the rifle was powered by a stiff spring instead of an explosion we would have got the same result.

Objects colliding along a straight line

Objects colliding along a straight line Conservation of momentum in the x-direction gives:

Objects colliding along a straight line Conservation of momentum in the x-direction gives: Same momentum after the collision, so:

Objects colliding along a straight line Conservation of momentum in the x-direction gives:

Objects colliding along a straight line Conservation of momentum in the x-direction gives: Same momentum after the collision, so:

Objects colliding along a straight line

Now, consider a two-dimensional collision

Now, consider a two-dimensional collision Conservation of momentum in x & y directions gives:

Now, consider a two-dimensional collision

Elastic compared to inelastic

Completely (or nearly) inelastic collisions

Completely (or nearly) inelastic collisions Cars are designed to crumple and absorb as much energy as possible so the passengers do not need to.

The ballistic pendulum A ballistic pendulum is a system for measuring the speed of a bullet. This can be demonstrated with a 4 4 block and a.22 caliber rifle. The bullet of mass mb, is fired into a wood block of mass mw, suspended like a pendulum and makes a completely inelastic collision with it. After the bullet s impact the block swings up to a maximum height, y. If we know the values of y, mb, and mw, what is the initial speed of the bullet vi?

The ballistic pendulum

The ballistic pendulum Stage 1) Conservation of momentum

The ballistic pendulum Stage 1) Conservation of momentum Stage 2) Conservation of energy

The ballistic pendulum Stage 1) Conservation of momentum gives:

The ballistic pendulum Stage 1) Conservation of momentum gives: Stage 2) Conservation of energy gives: Initial kinetic energy Final potential energy

Classifying Collisions

A possible simple model for automobile accidents

A possible simple model for automobile accidents Components of momentum before the collision An Inelastic Collision Momentum Conserved Kinetic Energy NOT conserved

A possible simple model for automobile accidents Components of momentum before the collision An Inelastic Collision Momentum Conserved Kinetic Energy NOT conserved By conservation of momentum during the collision

Elastic collisions Billiard balls are a very good example of objects that collide elastically.

Q8.1 A ball (mass 0.40 kg) is initially moving to the left at 30 m/s. After hitting the wall, the ball is moving to the right at 20 m/s. What is the impulse of the net force on the ball during its collision with the wall? A. 20 kg m/s to the right B. 20 kg m/s to the left C. 4.0 kg m/s to the right D. 4.0 kg m/s to the left E. none of the above

A8.1 A ball (mass 0.40 kg) is initially moving to the left at 30 m/s. After hitting the wall, the ball is moving to the right at 20 m/s. What is the impulse of the net force on the ball during its collision with the wall? J = p p 2 1 A. 20 kg m/s to the right B. 20 kg m/s to the left C. 4.0 kg m/s to the right D. 4.0 kg m/s to the left E. none of the above = 0.40 20 0.40 30 = 20

Q8.2 You are testing a new car using crash test dummies. Consider two ways to slow the car from 90 km/h (56 mi/h) to a complete stop: (i) You let the car slam into a wall, bringing it to a sudden stop. (ii) You let the car plow into a giant tub of gelatin so that it comes to a gradual halt. In which case is there a greater impulse of the net force on the car? A. in case (i) B. in case (ii) C. The impulse is the same in both cases. D. not enough information given to decide

A8.2 You are testing a new car using crash test dummies. Consider two ways to slow the car from 90 km/h (56 mi/h) to a complete stop: (i) You let the car slam into a wall, bringing it to a sudden stop. (ii) You let the car plow into a giant tub of gelatin so that it comes to a gradual halt. In which case is there a greater impulse of the net force on the car? A. in case (i) B. in case (ii) C. The impulse is the same in both cases. D. not enough information given to decide J = p 2 p 1, in both cases the car comes to a complete stop so both p 1 & p 2 are the same

Q8.3 A 3.00-kg rifle fires a 0.00500-kg bullet at a speed of 300 m/s. Which force is greater in magnitude: (i) the force that the rifle exerts on the bullet; or (ii) the force that the bullet exerts on the rifle? A. the force that the rifle exerts on the bullet B. the force that the bullet exerts on the rifle C. both forces have the same magnitude D. not enough information given to decide

A8.3 A 3.00-kg rifle fires a 0.00500-kg bullet at a speed of 300 m/s. Which force is greater in magnitude: (i) the force that the rifle exerts on the bullet; or (ii) the force that the bullet exerts on the rifle? A. the force that the rifle exerts on the bullet B. the force that the bullet exerts on the rifle C. both forces have the same magnitude D. not enough information given to decide Newton s third law, for every action there is an equal and opposite reaction

Q8.4 Two objects with different masses collide and stick to each other. Compared to before the collision, the system of two objects after the collision has A B A. the same total momentum and the same total kinetic energy. B. the same total momentum but less total kinetic energy. C. less total momentum but the same total kinetic energy. D. less total momentum and less total kinetic energy. E. not enough information given to decide

A8.4 Two objects with different masses collide and stick to each other. Compared to before the collision, the system of two objects after the collision has A B A. the same total momentum and the same total kinetic energy. B. the same total momentum but less total kinetic energy. C. less total momentum but the same total kinetic energy. D. less total momentum and less total kinetic energy. E. not enough information given to decide An Inelastic Collision Momentum Conserved Kinetic Energy NOT conserved

Q8.5 Two objects with different masses collide and bounce off each other. Compared to before the collision, the system of two objects after the collision has A B A. the same total momentum and the same total kinetic energy. B. the same total momentum but less total kinetic energy. C. less total momentum but the same total kinetic energy. D. less total momentum and less total kinetic energy. E. not enough information given to decide

A8.5 Two objects with different masses collide and bounce off each other. Compared to before the collision, the system of two objects after the collision has A B A. the same total momentum and the same total kinetic energy. B. the same total momentum but less total kinetic energy. C. less total momentum but the same total kinetic energy. D. less total momentum and less total kinetic energy. E. not enough information given to decide Do not know if elastic or inelastic collision

Q8.6 Block A has mass 1.00 kg and block B has mass 3.00 kg. The blocks collide and stick together on a level, frictionless surface. After the collision, the kinetic energy (KE) of block A is A. 1/9 the KE of block B. B. 1/3 the KE of block B. C. 3 times the KE of block B. D. 9 times the KE of block B. E. the same as the KE of block B.

A8.6 Block A has mass 1.00 kg and block B has mass 3.00 kg. The blocks collide and stick together on a level, frictionless surface. After the collision, the kinetic energy (KE) of block A is A. 1/9 the KE of block B. B. 1/3 the KE of block B. C. 3 times the KE of block B. D. 9 times the KE of block B. E. the same as the KE of block B. The ratio of the two kinetic energies is equal to the inverse ratio of their masses.

Q8.7 Block A on the left has mass 1.00 kg. Block B on the right has mass 3.00 kg. The blocks are forced together, compressing the spring. Then the system is released from rest on a level, frictionless surface. After the blocks are released, the kinetic energy (KE) of block A is A. 1/9 the KE of block B. B. 1/3 the KE of block B. C. 3 times the KE of block B. D. 9 times the KE of block B. E. the same as the KE of block B.

A8.7 Block A on the left has mass 1.00 kg. Block B on the right has mass 3.00 kg. The blocks are forced together, compressing the spring. Then the system is released from rest on a level, frictionless surface. After the blocks are released, the kinetic energy (KE) of block A is A. 1/9 the KE of block B. B. 1/3 the KE of block B. C. 3 times the KE of block B. D. 9 times the KE of block B. E. the same as the KE of block B. Conservation of momentum

Q8.8 An open cart is rolling to the left on a horizontal surface. A package slides down a chute and lands in the cart. Which quantities have the same value just before and just after the package lands in the cart? A. the horizontal component of total momentum B. the vertical component of total momentum C. the total kinetic energy D. two of A., B., and C. E. all of A., B., and C.

A8.8 An open cart is rolling to the left on a horizontal surface. A package slides down a chute and lands in the cart. Which quantities have the same value just before and just after the package lands in the cart? A. the horizontal component of total momentum B. the vertical component of total momentum C. the total kinetic energy D. two of A., B., and C. E. all of A., B., and C.

Angular Momentum Every rotational quantity we have encountered has a translational analog. The analog of momentum, p=mv, is angular momentum,

Angular Momentum

Conservation of Angular Momentum The conservation of angular momentum follows directly from As τ = d L dt If τ = 0 then d L dt = 0, and so L remains constant

A Falling Cat

The professor as ballerina? Angular momentum is conserved.

This is how a car s clutch works!

This is how a car s clutch works! This is analogous to a completely inelastic collision

Q10.11 A spinning figure skater pulls his arms in as he rotates on the ice. As he pulls his arms in, what happens to his angular momentum L and kinetic energy K? A. L and K both increase. B. L stays the same, K increases. C. L increases, K stays the same. D. L and K both stay the same.

A10.11 A spinning figure skater pulls his arms in as he rotates on the ice. As he pulls his arms in, what happens to his angular momentum L and kinetic energy K? A. L and K both increase. B. L stays the same, K increases. C. L increases, K stays the same. D. L and K both stay the same.