Work, Energy, Power

Chapter 7 in a nutshell

Work is Force times Distance.The change in Kinetic Energy is equal to the work.

Power is Work per unit time.

TWP

KW

dFW

/

New Concept:Kinetic Energy

K = ½ m V2

Work causes a change in kinetic energy, so the units are the same.

Units of Kinetic energy: Joule

But, from the definition of kinetic energy, the units are also,

1 Joule = 1 Kg-m2/s2

Table 7-1Typical Values of Work

Annual U.S. energy use 8 x 1019

Mt. St. Helens eruption 1018

Burning one gallon of gas 108

Human food intake/day

Melting an ice cube

107

104

Lighting a 100-W bulb for 1 minute 6000

Heartbeat 0.5

Turning page of a book 10–3

Hop of a flea 10–7

Breaking a bond in DNA 10–20

Activity Equivalent work (J)

Work is force times distance…but!

Only the force component in the direction of motion counts!

Units of work: Joule

Force in direction of motion is what matters…

D

F

Fx = F cos

W = D * F cos

Figure 7-3Force at an Angle to Direction of Motion: Another Look

I’ve got work to do….

F = Mg

F = Mg

y

W=F*y=MgH

This is the work done by the person lifting the box.

How about the reverse?….

F = Mg

F = Mg

H

W=F*y = - MgH

Work can be positive or negative.Lifting a box is positive work.Lowering the box is negative work.

Positive or Negative?Work done by gravity.

Fg = - Mg

y = - H

Work “done by force of gravity” is positive when an object is dropped.

Problem 7-77

W = 50J, given v, F, x.

What is theta?What is M?

1

1

(cos )

cos

cos

50.0 Jcos

(45.0 N)(1.50 m)

42.2

W F dW

FdW

Fd

2 2 2f i f

1 1( )

2 2W K m v v mv

2 2mfs

2 2(50.0 J)14.8 kg

2.60

Wm

v

Figure 7-4Positive, Negative, and Zero Work

Raindrops are falling on my head….

V1

F = - Mg

V2

y = -H

MgH

HMg

yFW

)()(

21

22 2

1

2

1MvMv

KW

)(2

1 21

22 vvMMgH

Special case: Object dropped from rest.

-Mg

Mass M dropped from height H. What is speed just before hitting the ground? (Neglect friction of air)

H

W=MgH

W=K=1/2 M v2

gHv 2

Thought provoking: The inclined plane.

Distance = hForce = mgW = mgh

Distance = L = h/sin Force = mg sin W = mgh

The work is the same in both cases. This is an example of conservation of energy, which we will see much more of in the future.

How about the speed?

H

If Block fell straight down through height H, its speed would be:

gHv 2

Use F=Ma to find speed down the ramp.

F=Mg sin

V=atL=1/2 a t2 so

aL

a

Lav

2

2

But a = g sinAnd L = H/sin

So…

gHv 2

Same magnitude of speed, but DIFFERENT DIRECTION!

Figure 7-6Graphical Representation of the Work Done

by a Constant Force

Figure 7-7Work Done by a Non-Constant Force

Figure 7-8aWork Done by a Continuously Varying Force

Figure 7-8bWork Done by a Continuously Varying Force

Figure 7-8cWork Done by a Continuously Varying Force

When force is not constant.The area under a force-distance

curve is the work, W.

Distance

For

ce

The average force times the distance gives the work. This is the same as the area under the curve.

Back to our Spring Flings.

F = -K x

How much work is done to move the block?Be careful with the signs of the forces!The force to move the block must be equal and opposite to the force of the spring on the block.

F = K x

For

ce

Distance X

F=Kx

W = ½ F XW = ½ K X2

Work done to move a mass on a spring a distance “x”:

221 KxW

Conceptual Checkpoints

• What is work, in physics terms?

• What is kinetic energy?

• What is power?

Work is force in the direction of motion.

Kinetic energy is ½ Mv2

Power is Work (or Energy) per Unit time.

Work and the WeightlifterA weightlifter performs three steps. What is the work done for each of the steps? Assume the weight is 100 Kg, the height of the lift is 1 M, and the acceleration of gravity is 10 M/s^2.

Work done against gravity to lift an object a height H is W = MgH

W1: The weight is lifted up to the height of 1 M.W2: The weight is held at 1M for 5 seconds.W3: The weight is lowered down to the ground.

Cheryl Hawarth, USA gold medal.

Weightlifting Work.

1. 1000J, 5000J, 1000J2. 1000J, 0J, -1000J3. 1000J, 5000J, -1000J

W1: The weight is lifted up to the height of 1 M.W2: The weight is held at 1M for 5 seconds.W3: The weight is lowered down to the ground.

The work done W1, W2, W3 is:

The energy is kinetic, relatively speaking….

V = + 5 m/s

V = -5 m/s

Train moves to the right with speed V. A baseball is thrown to the left with the same speed.

What is the kinetic energy of the baseball? Its mass is 1 kg.

What is the kinetic energy?

1. 12.5 J2. 5 J3. Zero4. Either zero or 12.5J,

depends on your frame of reference.

V = + 5 m/s

V = -5 m/s

K = ½ M V2

M = 1 kg

The Power to overcome Friction.Work is Force times distance. Power is Work per unit time.

F

Frictional Force, Ff = N

Block moves with STEADY SPEED under influence of applied force that EXACTLY balances friction, so V is CONSTANT.

AN IMPORTANT SPECIAL CASE

Force of friction Under these conditions,

vFT

XF

T

WP

THIS IS A SPECIAL CASE IN WHICH AN APPLIED FORCE RESULTS IN CONSTANT SPEED.IT OCCURS BECAUSE THE NET FORCE IS ZERO (SUM OF ALL APPLIED FORCES).

How about this case?

Total work is just the sum of the work for each spring.

221

22

21

2

12

1

2

1

xKK

xKxKW

And this one?This is harder. The two springs DO NOT stretch the same distance (they have different K values).

2

2

1

1

222

111

21

k

F

k

Fx

xkF

xkF

xxx

But F1 = F2 (why?), so let it be called F.

21

11

kkFx

So, can create an effective spring constant, Keff, 1

21

21

1111

1

kkkk

keff

Then, we can apply the usual formula: 2

2

1xkW eff