6. Work, Energy & Power

1. Work

2. Forces that Vary

3. Kinetic Energy

4. Power

What’s the speed of these skiers at the bottom of the slope?

Does the work of climbing a mountain depend on the route chosen?

No

Direct application of Newton’s law can be infeasible.

simple complicatedEnergy conservation to the rescue (Chap 7).

6.1. Work

Work W done on an object by a constant force F is

FW F r rF = displacement along direction of F.

Note: F need not be a net force.

1W Joule N m

cosW F x

FF x

Fr = force along direction of r .rF r

Example 6.1. Pushing a Car

The man pushes with a force of 650 N, moving the car 4.3 m.

How much work he does?

W F x 650 4.3N m 2.8 kJ

Example 6.2. Pulling a Suitcase

Woman exerts 60 N force on suitcase, pulling at 35 angle to the horizontal.

How much work is done if the suitcase is moved 45 m on a level floor?

xW F x

60 cos 35 45N m

2.2 kJ

x

xF = x cos

FF x

cosF x

cosF x

Work & the Scalar Product

Work is a scalar.

Scalar = quantity specified by a single number that is the same in every coordinate system.

Scalar has no direction.

Scalar (dot) product of vectors A & B :

cosA B A B is a scalar

cosW F r = angle between F & r

W F r

x x y yA B A B A

B

BA = B cos

cosA B A BAA B

BA B

x x y y z zA B A B A B 3-D

2-D

Example 6.3. Tugboat

Tug boat pushes a cruiser with force F = ( 1.2, 2.3 ) MN,

displacing the ship by r = ( 380, 460 ) m.

(a)Find the work done by the tugboat.

(b) Find the angle between F & r.

W F rx yF x F y

1.2 380 2.3 460MN m MN m

1510 MJ

cosF r

2 2x yF F F 2 2

1.2 2.3MN MN 2.59 MN

2 2r x y 2 2

380 460m m 597 m

1cosW

F r

1 1510

cos2.59 597

MJ

MN m 12

6.2. Forces that Vary

1

N

ii

W W

1

N

ii

F X x

1

limN

iN

i

W F X x

2

1

x

xW F x d x

2 1x xx

N

1

1

2iX x i x

1 1

1

2X x x

2 11

1

2N

x xX x N

N

2

1

2x x

Tactics 6.1. Integrating

inverseDerivative Indefinite Integral Antiderivative

1n

nd xn x

d x

ng x d x nd gx

d x

g f d x

d g

fd x

Example:

Since we have1

1

nxg

n

See Appendix A for integral table.

2 2

11

x xn

xxx d x g

1 12 1

1 1

n nx x

n n

2 1g x g x

1

1

nn x

x dxn

Stretching a Spring

0

xW F x d x 0

xk x d x

2

0

1

2

x

k x1

1

nn x

x dxn

21

2k x

Example 6.4. Bungee Jumping

Bungee cord is 20 m long with k = 11 N/m.

At lowest point, cord length is doubled.

(a) How much work is done on cord?

(b) How does work done in the last meter compare with that done in the 1st meter?

2 21 2 2 1

1

2W k x x

(a) 2 2111 / 20 0

2W N m m m 2.2 kJ

(b) 1st meter

2 2111 / 1 0

2W N m m m 5.5 J

Last meter

2 2111 / 20 19

2W N m m m 214 J

Example 6.5. Rough Sliding

Workers pushing a 180 kg trunk across a level floor encounter a 10 m region where floor

becomes increasingly rough.

There, k = 0 + a x2, with 0 = 0.17, a = 0.0062 m2 & x is the distance into the rough part.

How much work does it take to push the trunk across the region?

2

1

x

xW F x d x kF x m g

2

1

20

x

xW ax m g d x

2

1

30

1

3

x

x

m g x a x

3 30 2 2 0 1 1

1 1

3 3m g x a x x a x

32 21180 9.8 / 0.17 10 0.0062 10

3kg m s m m m

6.6 kJ

1

1

nn x

x dxn

Force & Work in 2- & 3- D

2

1

W dr

rF r r Line integral

2

1x yF d x F d y

r

r

2

1x y zF d x F d y F d z

r

r

2

1

x

xF d x 1-D

2-D

3-D

Work Done Against Gravity

Only vertical displacement requires work against gravity

W = m g h

W m g y

GOT IT? 6.2.

3 forces have magnitudes in N that are numerically equal to

(a) x, (b) x2, (c) x,

where x is the position in meters.

Each force moves an object from x = 0 to x = 1 m.

Note that each force has the same values at the end points, namely, 0 N & 1 N.

Which force does the most work?

Which does the least?

(c)

(b)

6.3. Kinetic Energy

net netW F d x dvm d x

dt

d xm dv

dt m v dv

2

1

v

net vW m v dv

2

1

21

2

v

v

m v 2 22 1

1 1

2 2m v m v

21

2m v

Kinetic energy: 21

2K m v

K is relative (depends on reference frame).

K is a scalar.

Work-energy theorem: netK W

Example 6.6. Passing Zone

A1400 kg car enters a passing zone & accelerates from 70 to 95 km/h.

(a) How much work is done on the car?

(b) If the car then brakes to stop, how much work is done on it?

2 22 1

1 1

2 2netW K m v m v

(a) 2 211400 95 / 70 /

2netW kg km h km h 22,887,500 /kg km h

21000

28875003600

mkg

s

222.8 kJ

b) 2 211400 0 95 /

2netW kg km h 26,317,500 /kg km h

21000

6,317,5003600

mkg

s

487.5 kJ

GOT IT? 6.3.

For each situation, tell whether the net work done on a soccer ball is

positive, negative, or zero.

Justify your answer using the work-energy theorem.

(a)You carry the ball to the field, walking at constant speed.

(b) You kick the stationary ball, starting it flying through the air.

(c) The ball rolls along the filed, gradually coming to a halt.

zero (K=0)

positive (K>0)

negative (K<0)

Energy Units

[energy] = [work] = J (SI)

CGS: 21 1 /erg g cm s 710 J

Other energy units:

eV (electron-volt): used in nuclear, atomic, molecular, solid state physics.

cal (calorie), BTU (British Thermal Unit): used in thermodynamics.

kW-h (kilowatt-hours): used in engineering.

See Appendix C

6.4. Power

Average power:W

Pt

(Instantaneous) power:0

limt

WP

t

dW

d t

power watt /W J s

Example 6.7. Climbing Mount Washington

A 55 kg hiker makes the vertical rise of 1300 m in 2 h.

A 1500 kg car takes ½ h to go there.

Neglecting loss to friction, what is the average power output for each.

Hiker:m g hP

t

255 9.8 / 1300

2 3600 /

kg m s mP

h s h 97W

21500 9.8 / 1300

0.5 3600 /

kg m s mP

h s hCar: 11 kW

W P t for constant power P

0limt

W P t

WP

t

2

1

t

tP dt general case

Example 6.8. Yankee Stadium

Each of the 500 floodlights at Yankee stadium uses 1.0 kW power.

How much do they cost for a 4 h night game, if electricity costs 9.5 ₵ / kW-h ?

W P t 500 1.0 4 $ 0.095 /kW h kW h $190

Energy and Society

2008: 4.71020 J or 1.51013 W

Power & Velocity

dW d F rd

dt F

rdWP

d t P F v

Example 6.9. Bicycling

Riding a 14 kg bicycle at a steady 18 km/h (5.0 m/s),

you experience a 30 N force from air resistance.

If you mass 68 kg, what power must you supply

(a)on level ground.

(b) up a 5 slope.

(a) airP F v 30 5.0 /N m s 150W

(b) air gP F F v sinairF m g v

230 14 68 9.8 / sin 5 50 /N kg kg m s m s 500W

Fair

Fg

vFair

v