Conservation of Energy and Energy Diagrams

Monday, October 24, 11

Summary

Monday, October 24, 11

Summary

Monday, October 24, 11

Summary

Monday, October 24, 11

Summary

Monday, October 24, 11

Summary

Monday, October 24, 11

The stretch of a spring and the force that caused it The force applied to an ideal spring will be proportional to its stretch. The graph of force on the y axis versus stretch on the x axis will yield a slope of k, the spring constant.

Monday, October 24, 11

Stepping on a scale Whether you like the result or not, stepping on a scale is an excellent example of applied force and the work being done to compress that spring.

Monday, October 24, 11

Motion with a varying force

Monday, October 24, 11

Motion with a varying force

Monday, October 24, 11

Motion on a curved path If you watch a child on a swing set, you can also consider the motion of a particle along a curved path.

Monday, October 24, 11

Watt about power? Once work is calculated, dividing by the time that passed determines power. The pun is credit to James Watt. (You will see that scientists of that era often were privileged to leave their names on the topic of their efforts.) Also note the popular culture power unit of horsepower. The energy you use may be noted from the meter the electric company probably installed to measure your consumption of energy in kilowatt-hours.

Monday, October 24, 11

Power

Monday, October 24, 11

Power

Monday, October 24, 11

Power • •

Same mass... Both reach 60 mph...

Same final kinetic energy, but different times mean different powers.

Monday, October 24, 11

An example you might do if the elevator is out It’s interesting how a lighter stair climber and heavier stair climber can expend the same power by using different amounts of time.

Monday, October 24, 11

Summary

Monday, October 24, 11

Summary

Monday, October 24, 11

Gravitational potential energy, U Energy associated with position

The path of the basketball can demonstrate many principles of potential and kinetic energy. Monday, October 24, 11

Can express this as a function of gravitational potential energy, U

So the work done by the gravitational force during the displacement from y1 to y2 is:

Monday, October 24, 11

The sign of ΔU

Monday, October 24, 11

The sign of ΔU

Monday, October 24, 11

Total Mechanical Energy of the System

Suppose that a body is falling freely with no air resistance, then the body’s weight is the only force acting. If its speed at point y1 is v1, and its speed at y2 is v2, then the work-energy theorem gives us Wtot = ΔK = K 2 − K1 Wtot = Wgrav = −ΔU grav = U grav,1 − U grav,2 so ΔK = −ΔU grav or K 2 − K1 = U grav,1 − U grav,2

Which we can rewrite as:

Monday, October 24, 11

Total Mechanical Energy of the System

The total mechanical energy of the system is called E

E = K + U grav A quantity that always has the same value is called a conserved quantity. When only the force of gravity does work, the total mechanical energy is conserved.

Monday, October 24, 11

Athletes and the conservation of energy Notice how potential and kinetic energy interchange as the athlete jumps.

Monday, October 24, 11

Zero Height

Monday, October 24, 11

Athletes and energy Notice how velocity changes as forms of energy interchange.

Monday, October 24, 11

Height of a Baseball from Energy Conservation You throw a 0.145 kg baseball straight up in the air, giving it an initial upward velocity of magnitude 20.0 m/s. Find how high it goes, ignoring air resistance.

Monday, October 24, 11

Height of a Baseball from Energy Conservation You throw a 0.145 kg baseball straight up in the air, giving it an initial upward velocity of magnitude 20.0 m/s. Find how high it goes, ignoring air resistance.

y2

y1=0

Monday, October 24, 11

Height of a Baseball from Energy Conservation You throw a 0.145 kg baseball straight up in the air, giving it an initial upward velocity of magnitude 20.0 m/s. Find how high it goes, ignoring air resistance.

y2

y1=0

Monday, October 24, 11

Height of a Baseball from Energy Conservation You throw a 0.145 kg baseball straight up in the air, giving it an initial upward velocity of magnitude 20.0 m/s. Find how high it goes, ignoring air resistance.

y2

y1=0

Monday, October 24, 11

Height of a Baseball from Energy Conservation You throw a 0.145 kg baseball straight up in the air, giving it an initial upward velocity of magnitude 20.0 m/s. Find how high it goes, ignoring air resistance.

y1 = 0 ⇒ U grav,1 = mgy1 = 0

y2

1 2 v2 = 0 ⇒ K 2 = mv2 = 0 2 K1 = U grav,2

y1=0

Monday, October 24, 11

Height of a Baseball from Energy Conservation

y2

y1 = 0 ⇒ U grav,1 = mgy1 = 0 1 2 v2 = 0 ⇒ K 2 = mv2 = 0 2 K1 = U grav,2 1 2 1 K1 = mv1 = × 0.145kg × 20.0m / s = 29.0J 2 2 U grav,2 29.0J U grav,2 = mgy2 ⇒ y2 = = = 20.4m 2 mg 0.145kg × 9.80m / s

Monday, October 24, 11

y1=0

Forces other than gravity doing work We already know that: If we have forces additional to gravity at work then we have:

Wtot = Wother + Wgrav = K 2 − K1 = ΔK As we also know that:

Wgrav = U grav,1 − U grav,2 = −ΔU grav So substituting and rearranging:

K1 + U grav,1 + Wother = K 2 + U grav,2 Substituting the appropriate energy expressions we get

1 2 1 2 mv1 + mgy1 + Wother = mv2 + mgy2 2 2 Monday, October 24, 11

Forces other than gravity doing work 1 2 1 2 mv1 + mgy1 + Wother = mv2 + mgy2 2 2 When we are considering a problem using mechanical energy it is often best to use the energy approach if we have varying forces, and/ or motion along a curved path.

Monday, October 24, 11

Forces other than gravity doing work You throw a 0.145 kg baseball straight up in the air, and while you are throwing the ball your hand moves up 0.50 m. The ball leaves your hand with an initial upward velocity of magnitude 20.0 m/s. Ignore air resistance. a) assuming your hand exerts a constant upward force on the ball, find the magnitude of that force. b) Find the speed of the ball at a point 15 m above the point that it leaves your hand. Monday, October 24, 11

Forces other than gravity doing work a)

y1 = −0.50m ⇒ U grav,1 = 0.145kg × 9.80ms / s 2 × −0.50m = −0.71J v1 = 0 ⇒ K1 = 0 y2 = 0 ⇒ U grav,2 = 0 1 2 v2 = 20m / s ⇒ K 2 = 0.145kg × (20.0m / s) = 29.0J 2

Monday, October 24, 11

Forces other than gravity doing work a)

y1 = −0.50m ⇒ U grav,1 = 0.145kg × 9.80ms / s 2 × −0.50m = −0.71J v1 = 0 ⇒ K1 = 0 y2 = 0 ⇒ U grav,2 = 0 1 2 v2 = 20m / s ⇒ K 2 = 0.145kg × (20.0m / s) = 29.0J 2

Wother = (K 2 − K1 ) + (U grav,2 − U grav,1 ) = (29.0J − 0) + (0 − (−0.71J )) = 29.7J = F(y2 − y1 ) Wother 29.7J F= = = 59N (y2 − y1 ) 0.50m

Monday, October 24, 11

Forces other than gravity doing work b) Note that between points 2 and 3 the force of your hand no longer acts so mechanical energy is conserved as Wother=0

K 2 + U grav,2 = K 3 + U grav, 3 U grav, 3 = mgy3 = 0.145kg × 9.80m / s × 15.0m = 21.3J 2

K 3 = (K 2 + U grav,2 ) − U grav, 3 = (29.0J + 0) − 21.3J = 7.7J 1 2 2K 3 2 × 7.7J = mv3y ⇒ v3y = ± =± = ±10m / s 2 m 0.145kg Note that the ball passes point 3 twice, on the way up, and on the way down.

Monday, October 24, 11

Work and energy along a curved path Notice the same expression for gravitational potential energy is the same along either a curved  or straight path. ii j = 0      W = FiΔs = −mg ji(Δx i + Δy j) = −mgΔy Why the minus? So the total work is:

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Consider projectile motion using energetics

Monday, October 24, 11

What’s the speed in a vertical circle?

Monday, October 24, 11

What’s the speed in a vertical circle?

Monday, October 24, 11

What’s the speed in a vertical circle?

Monday, October 24, 11

What’s the speed in a vertical circle?

Monday, October 24, 11

Speed in a vertical circle with friction Suppose that the ramp was not frictionless and his speed at the bottom was only 6.00m/s. What work was done by the frictional force acting on him?

Monday, October 24, 11

Speed in a vertical circle with friction Suppose that the ramp was not frictionless and his speed at the bottom was only 6.00m/s. What work was done by the frictional force acting on him? Wother = (K 2 − K1 ) + (U grav,2 − U grav,1 )

Monday, October 24, 11

Speed in a vertical circle with friction Suppose that the ramp was not frictionless and his speed at the bottom was only 6.00m/s. What work was done by the frictional force acting on him? Wother = (K 2 − K1 ) + (U grav,2 − U grav,1 )

Monday, October 24, 11

Elastic Potential Energy

Monday, October 24, 11

Elastic Potential Energy For an ideal spring we know that the magnitude of the force exerted to stretch it a distance x is:

F = kx

Where k is the force constant of the spring. X

X

1 2 W = ∫ Fx dx = ∫ kx dx = kX 2 0 0

Monday, October 24, 11

Elastic Potential Energy So the work done on a spring to move it from elongation x1 to x2 is

From Newton’s third law, the work done by a spring from elongation x1 to x2 is

“el” stands for elastic

Monday, October 24, 11

Elastic Potential Energy

Monday, October 24, 11

Work and energy in the motion of a mass on a spring

Monday, October 24, 11

Work and energy in the motion of a mass on a spring

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Monday, October 24, 11

Monday, October 24, 11

“Bungee jumping” requires we look at two potential energies In this case we need to consider gravitational potential energy, kinetic energy, and elastic potential energy.

Monday, October 24, 11

Motion with elastic potential energy

Monday, October 24, 11

Motion with elastic potential energy

K 2 − K1 = U1 − U 2 ⇒ K 2 = K1 + U1 − U 2

K 2 = K1 + U1 − U 2 = 0 + 0.0250J − 0.0160J = 0.0090J 2K 2 2 × 0.0090J v2 x = ± =± = ±0.30m / s m 0.200kg Monday, October 24, 11

Bring together two potential energies and friction

The fall of the elevator is stopped by a spring and by a constant friction force. Monday, October 24, 11

Bring together two potential energies and friction The elevator’s initial speed, v1, is 4.00m/s so the initial kinetic energy, K1, is

Monday, October 24, 11

Bring together two potential energies and friction

Monday, October 24, 11

Conservative forces The work by a conservative force like gravity does not depend on the path your hiking team chooses, only how high you climb.

Monday, October 24, 11

Conservative forces

Monday, October 24, 11

Does Friction Depend on the Path Taken?

Monday, October 24, 11

Friction does depend on the path taken Nonconservative frictional force changes with path.

Monday, October 24, 11

Friction does depend on the path taken Nonconservative frictional force changes with path.

The magnitude of the frictional force is:

fk = µ k n = µ k mg

Monday, October 24, 11

Friction does depend on the path taken Nonconservative frictional force changes with path. The magnitude of the frictional force is: fk = µ k n = µ k mg

Monday, October 24, 11

The law of Conservation of Energy

Monday, October 24, 11

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.

ΔU Fx (x)Δx = −ΔU ⇒ Fx (x) = − Δx So in the limit as Δx → 0 dU(x) Fx (x) = − dx Monday, October 24, 11

Force as the derivative of potential energy ΔU Fx (x)Δx = −ΔU ⇒ Fx (x) = − Δx So in the limit as Δx → 0 dU(x) Fx (x) = − dx

Monday, October 24, 11

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.

Monday, October 24, 11

The potential energy curve for motion of a particle

dU Fx = − dx Monday, October 24, 11

Energy diagrams give us insight

Monday, October 24, 11

Summary

Monday, October 24, 11

Summary

Monday, October 24, 11

Summary

Monday, October 24, 11

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.

Monday, October 24, 11

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.

Monday, October 24, 11

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 y2. What is the speed of the ball when it is at a height of y2/2? Ignore air resistance. A. 10.0 m/s

v2 = 0

v1 = 20.0 m/s m = 0.150 kg

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

Monday, October 24, 11

y2

y1 = 0

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 y2. What is the speed of the ball when it is at a height of y2/2? Ignore air resistance. A. 10.0 m/s

v2 = 0

v1 = 20.0 m/s m = 0.150 kg

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

Monday, October 24, 11

y2

y1 = 0

Q7.4 The two ramps shown are both frictionless. The heights y1 and y2 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.

Monday, October 24, 11

A7.4 The two ramps shown are both frictionless. The heights y1 and y2 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.

Monday, October 24, 11

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 Ugrav and the elastic potential energy Uel? A. Ugrav and Uel are both increasing. B. Ugrav and Uel are both decreasing. C. Ugrav is increasing, Uel is decreasing. D. Ugrav is decreasing, Uel is increasing. E. The answer depends on how the block’s speed is changing.

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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 Ugrav and the elastic potential energy Uel? A. Ugrav and Uel are both increasing. B. Ugrav and Uel are both decreasing. C. Ugrav is increasing, Uel is decreasing. D. Ugrav is decreasing, Uel is increasing. E. The answer depends on how the block’s speed is changing.

Monday, October 24, 11

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 xdirection. 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

E. more than one of the above

Monday, October 24, 11

D. at x = d

A7.6 The graph shows the potential energy U for a particle that moves along the x-axis.

Potential energy lowest, kinetic energy highest

The particle is initially at x = d and moves in the negative xdirection. 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

E. more than one of the above

Monday, October 24, 11

D. at x = d

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 xdirection. At which of the labeled x-coordinates is the particle slowing down?

A. at x = a

B. at x = b

C. at x = c

E. more than one of the above

Monday, October 24, 11

D. at x = d

A7.7 The graph shows the potential energy U for a particle that moves along the x-axis.

The slope is positive so the acceleration is negative

The particle is initially at x = d and moves in the negative xdirection. At which of the labeled x-coordinates is the particle slowing down?

A. at x = a

B. at x = b

C. at x = c

E. more than one of the above

Monday, October 24, 11

dU Fx = − = ma x dx

D. at x = d

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.

Monday, October 24, 11

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

dU Fx = − 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.

Monday, October 24, 11

Q7.9 The graph shows a conservative force Fx as a function of x in the vicinity of x = a. As the graph shows, Fx = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct?

Fx

0

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.

Monday, October 24, 11

x a

A7.9 The graph shows a conservative force Fx as a function of x in the vicinity of x = a. As the graph shows, Fx = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct?

Fx

0

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.

Monday, October 24, 11

x a

Monday, October 24, 11

Q7.10 The graph shows a conservative force Fx as a function of x in the vicinity of x = a. As the graph shows, Fx = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct?

Fx

0

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.

Monday, October 24, 11

x a

A7.10 The graph shows a conservative force Fx as a function of x in the vicinity of x = a. As the graph shows, Fx = 0 at x = a. Which statement about the associated potential energy function U at x = a is correct?

Fx

0

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.

Monday, October 24, 11

x a

Monday, October 24, 11

Q7.11 The graph shows a conservative force Fx as a function of x in the vicinity of x = a. As the graph shows, Fx > 0 and dFx/dx < 0 at x = a. Which statement about the associated potential energy function U at x = a is correct? 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.

Monday, October 24, 11

Fx

0

x a

A7.11 The graph shows a conservative force Fx as a function of x in the vicinity of x = a. As the graph shows, Fx > 0 and dFx/dx < 0 at x = a. Which statement about the associated potential energy function U at x = a is correct?

Fx

0

x a

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.

dU Fx = − dx At a Fx>0 so dU/dx