Work, Energy, and Power

§ 6.2–6.4

Kinetic Energy

Energy of a moving mass

§ 6.2

CPS Question

To accelerate an object from 10 to 20 m/s requires

A. more work than to accelerate from 0 to 10 m/s.

B. the same amount of work as to accelerate from 0 to 10 m/s.

C. less work than to accelerate from 0 to 10 m/s.

Work of Acceleration• To accelerate to speed v with constant force F

t

v

speed

time

mvtF = m (slope) =

12 d = vt

• Work = F·d

• Work = = mv212vt1

2mvt

slope = a = Fm

vt=

area = d

Another Perspective

• So, for the 0–10 vs. 10–20 m/s case:

• If same force, then same time– a’s and v’s are equal, so t’s are equal

• Average speeds are 5 vs. 15 m/s

• The 10–20 m/s case travels 3 as far

A Moving Object Can Do Work

Source: Griffith, The Physics of Everyday Phenomena

Kinetic Energy

the work to bring a motionless object to speed

K = 12 mv2

equivalent to

the work a moving object does in stopping

Which has more kinetic energy?

A.

B.

C.

D.

10 kg 10 m/s

5 kg 10 m/s

10 kg 20 m/s

40 m/s5 kg

CPS Question

Rebounding Ball

What is the sign of the work done on the ball by the wall as it slows (squishes)?

A. Positive (W > 0).

B. Negative (W < 0).

C. Zero (W = 0).

D. Can’t tell (W = ?).

Rebounding Ball

What is the sign of the work done on the ball by the wall as it rebounds (expands)?

A. Positive (W > 0).

B. Negative (W < 0).

C. Zero (W = 0).

D. Can’t tell (W = ?).

Happy/Sad Balls

Which ball has the greatest change in kinetic energy K during impact?

A. The happy (rebounding) ball.

B. The sad (dead) ball.

C. Both had the same K.

Happy/Sad Balls

Which ball has the most (largest absolute value) work done on it during impact?

A. The happy (rebounding) ball.

B. The sad (dead) ball.

C. Both had the same W.

Worksheet Problem 1

A luge and its rider, total mass 90 kg, emerges onto a level track with v0 = 36 m/s. It undergoes a constant deceleration of 2.0 m/s2 until it stops.

a) What is the magnitude of the force acting on it?

b) What distance does it travel while decelerating?i. First find the general kinematic formula for distance x

traveled in stopping from speed v0 at acceleration a.

ii. Then find x in this case.

c) What work does the force do?

Worksheet Problem 1

A luge and its rider, total mass 90 kg, emerges onto a level track with v0 = 36 m/s. It undergoes a constant deceleration of 4.0 m/s2 until it stops.

d) What is the magnitude of the force acting on it?

e) What distance does it travel while decelerating?– Use the general formula found earlier.

f) What work does the force do?

g) What was the initial kinetic energy K0 of the luge?

CPS Question

The piglet has a choice of three frictionless slides to descend. Along which slide would the piglet finish with the highest speed?

A B C

D. The final speed is the same for all.

Worksheet Problem 2

A piglet slides down a frictionless ramp of height H and angle above the horizontal. What is its speed vf at the end?

a) Find the general kinematic equation for vf when accelerating from rest at acceleration a through a distance x.

b) Geometrically find a and x in terms of H and .

c) Find vf in this case.

d) What is the dependency of vf on ?

CPS Question

The piglet has a choice of three frictionless slides to descend. Along which slide would the piglet finish soonest?

A B C

D. The time is the same for all.

CPS Question

Now the piglet/slide interface has a little friction. Along which slide would the piglet finish with the highest speed?

A B C

D. The final speed is the same for all.

Work-Energy Theorem

• If an amount of work W is done on an otherwise isolated system, the system’s energy changes by an amount E = W.

• The net work done on an object equals its change in kinetic energy Wtot = K.

Work in General

curving paths, changing forces

§ 6.3

What’s the point?

• What is work when force is not constant or the path is not straight?

Work in General

• For constant force, W = F·s.

• In general, dW = F·ds.

• So, W = F·ds.• (Sum of work done over each interval.)

• Path may not be straight.

• F may vary with position or time.

Elastic Force

Stretching and squishing

§ 6.3

Structure of Solids

• Atoms and molecules connected by chemical bonds

• Considerable force needed to deform

compression tension

Elasticity of Solids

Small deformations are proportional to force

small stretch larger stretch

Hooke’s Law: ut tensio, sic vis (as the pull, so the stretch)

Robert Hooke, 1635–1703

Hooke’s Law Formula

F = force exerted by the spring

k = spring constant; units: N/m; k > 0

x = displacement from equilibrium position

negative sign: force opposes distortion

F = –kx

CPS Question

forward

backward

Displacement

Spring’s Force

What direction of force is needed to hold the object (against the spring) at its plotted displacement?A. Forward (right).

B. Backward (left).

C. No force (zero).

D. Can’t tell.

forwardbackward

Work to Deform a Spring

• Push or pull a distance x from equilibrium

x

kxforce

displacement

kx212

=

slope = k

area = w

• Work = F·x ;12 F = kx

• Work = kx·x12

CPS Question

A spring with force constant k is stretched from x = 0 to x = D. What is the work done by the spring as it stretches?

A. 1/2 kD2.

B. –1/2 kD2.

C. 0.

D. It cannot be determined.

E. None of these.

CPS Question

Two springs, one with a spring constant k1 and the other with a spring constant k2 = 2 k1, are slowly stretched to the same final tension. Which spring has more work done on it?

A. The stiffer spring (k = 2 k1)

B. The softer spring (k = k1)

C. The same work was done on both.

Worksheet Problem 3

A 1.53-kg block is released from rest just atop a relaxed spring with k = 2.50 N/cm. The block compresses the spring 12.0 cm before momentarily stopping.

a) While the spring is compressing, what work is done on the block:i. by gravity?ii. by the spring?

b) What force does the spring finally exert?c) What is the block’s final acceleration?d) What is the block’s initial acceleration?

Centripetal Force

work of acceleration

§ 6.3

Group CPS Question

A toy of mass m moving at constant speed v in a circle of radius r has a constant magnitude of centripetal acceleration of v2/r. Its velocity reverses every half-cycle.

How much work does the centripetal force do on the toy every half-cycle?

A. mv2.

B. –mv2.

C. mv2.

D. None of these.

Kinetic Energy and Direction

• K Depends on speed• Direction of velocity is irrelevant• Changing direction only requires force, but

no work.• 1/2 mv2 = 1/2 mvv is a scalar

Net Force and Net Work

• Net force is nonzero if a body accelerates

• net work is nonzero if a body changes speed

• The net force must overlap with the displacement to do work!

Worksheet Problem 4

Your cousin Throckmorton, m = 20 kg, plays on a R = 1.5-m swing. What is the net work done on him as he swings from an angle = /6 from vertical down to  = 0?

a) What is the net force as a function of ? b) How far (total path length) does he travel?c) Set up the integral for the total work done on him.d) Evaluate the total work.

Power

how quickly work is done

§ 6.4

Power

Rate of doing work

E = change in energy ( = work)

t = time interval

Power =E

t=

w

t

Units of Power

= J/s = W = watt

P =E

t

W =kg m2

s2 s

kg m2

s3=

Energytime

Power

A different but equivalent formula

P =Wt

=t

F·s= F·v

v = velocitys = displacement

F = force

CPS Question

Dragging a box across a level floor against friction at 1 m/s requires a power of 20 W. How much power is required to drag the same box at 2 m/s?

A. 10 W.

B. 20 W.

C. 40 W.

D. 80 W.