Work and Energy

Sections

Work Done by a Constant Force

Work Done by a Variable Force

The Work–Energy Theorem: Kinetic Energy

Potential Energy

Conservation of Energy

Power

Work

The work done by a constant force is defined as the distance moved multiplied by the component of the force in the direction of displacement:

Unit of work is Joules (J)

θcosFdW

In (a), there is a force but no displacement: no work is done. In (b), the force is parallel to the displacement, and in (c) the force is at an angle to the displacement.

Work

Work

What is the correct unit of work expressed in SI units?

(A) kg m2/s2

(B) kg m2/s

(C) kg m/s2

(D) kg2 m/s2

A delivery clerk carries a 3.4 kg package from the ground to the fifth floor of an office building, a total height of 15 m. How much work is done by the clerk?

Work

510 J

How much work does the force of gravity do when a 2.5 kg object falls a distance of 3.5 m?

Work

88 J

A rope is used to pull a metal box 15.0 m across the floor. The rope is held at an angle of 46o with the floor and a force of 628 N is used. How much work does the force on the rope do?

Work

6.54 x 103 J

How much work did the movers do (horizontally) pushing a 160 kg crate 10.3 m across a rough floor without acceleration, if the effective coefficient of friction was 0.50?

Work

HF

mgFF kNkf

d cosFW H

mgFF kfH dmgW k θcos

m 10.3 cos0m/s 9.8kg 160 0.5 2W

J 8075W

Work

Can work be done on a system if there is no motion?

(A) Yes, if an outside force is provided.

(B) Yes, since motion is only relative.

(C) No, since a system which is not moving has no energy.

(D) No, because of the way work is defined.

m

h

m

mg

Wg = mgh

Object falls in a gravitational field

θcosdFW gg

og mghW 0cos

Work

Energy

Work

A 50 N object was lifted 2.0 m vertically and is being held there. How much work is being done in holding the box in this position?

(A) more than 100 J

(B) 100 J

(C) less than 100 J, but more than 0 J

(D) 0 J

Kinetic Energy and the Work Energy Theorem

The quantity is

(A) the kinetic energy of the object.

(B) the potential energy of the object.

(C) the work done on the object by the force.

(D) the power supplied to the object by the force.

221 mv

If the speed of an arrow is doubled, by what factor does its KEincrease?

Kinetic Energy and the Work Energy Theorem (Problem)

2mv

KE2

2v2m

KE2

x

2mv

4KE2

x KE4

Work done is equal to the change in the kinetic energy:

• If the net work is positive, the kinetic energy increases.

• If the net work is negative, the kinetic energy decreases.

ifnet KEKEW

Kinetic Energy and the Work Energy Theorem

Kinetic Energy and the Work Energy Theorem

How much work is required to accelerate an 1125 kg car from 10 m/s to 25 m/s? (work on board)

Wnet = 2.95 x 105 J

When an object is thrown upward.

Earth

Negative workdone by the

gravitationalforce

Positive workdone by the

gravitationalforce

Gravitational Potential Energy

Gravitational Potential Energy

An object can have potential energy by virtue of its position.

Familiar examples of potential energy:

• A wound-up spring

• A stretched elastic band

• An object at some height above the ground

We therefore define the gravitational potential energy:

Fext

mg

y1

y2

h

In raising a mass m to a height h, the work done by the external force is

0 where cosdFW extext

mgh W

yymg

ext

12

mghPEg

m

Gravitational Potential Energy

Gravitational Potential Energy

The quantity is

(A) the kinetic energy of the object.

(B) the gravitational potential energy of the object.

(C) the work done on the object by the force.

(D) the power supplied to the object by the force.

mgh

How high will a 1.85 kg rock go if thrown straight up by someone who does 80.0 J of work on it? Neglect air resistance.

Gravitational Potential Energy (Problem)

Gravity does -80 J while stopping the rock.

cosmgdcosdFW gg

o2g

180cosm/s 9.8 gk 85.1

J 80cosmg

Wd

m 41.4d

The sum of the changes in the kinetic energy and in the potential energy is zero – the kinetic and potential energy changes are equal but opposite in sign.

This allows us to define the total mechanical energy:

ffoo PEKEPEKE

PEKE

0

PEKEEnergy Mechanical Total

And its conservation:

Systems and Energy Conservation

If there is no friction, the speed of a roller coaster will depend only on its height compared to its starting height.

y

Systems and Energy Conservation

So the first hill must be the highest, unless the track has chains/magnets along the way.

All three of these balls have the same initial kinetic energy; as the change in potential energy is also the same for all three.

Systems and Energy Conservation

Their speeds just before they hit the bottom are the same as well.

hKEWg

2mv

mgh2

gh2v v

mgh2

2mv

Ball dropped from rest falls freely from a height h.Find its final speed.

Systems and Energy Conservation

Power

Power is the rate at which work is done –

The difference between walking and running up the stairs is power – the change in gravitational potential energy is the same.

In the SI system, the units of power are watts (W):

TimeWork

Power Average Time

dTransforme Energy

SecondJoule

1Watt1

Power is also needed for acceleration and for moving against the force of gravity.

The average power can be written in terms of the force and the average velocity:

v

FFR

d

tW

P t

Fd Fv

Power

A box that weighs 57.5 kg is lifted a distance of 20.0 m straight up by a rope. The job is done in 10.0 s. What power is developed in watts and kilowatts?

Power

1.15 x 103 W = 1.15 kW

An electric motor develops 65 kW of power as it lifts a loaded elevator 17.5 m in 35.0 s. How much force does the motor exert?

Work

1.3 x 105 N

A 1000 kg sports car accelerates from rest to 20 m/s in 5.0 s. What is the average power delivered by the engine?

Power (Problem)

timeenergy

P

time

KE

t

vvmP

2o

221

s 5 2

0m/s 20kg 1000 22

W40000P

Review

Work done by a constant force is the displacement times the component of force in the direction of the displacement.

Kinetic energy is the energy of motion.

Work–energy theorem: the net work done on an object is equal to the change in its kinetic energy.

Potential energy is the energy of position or configuration.

Review

The total energy of the universe, or of an isolated system, is conserved.

Total mechanical energy is the sum of kinetic and potential energy. It is conserved in a conservative system.

The net work done by forces is equal to the change in the total mechanical energy.

Power is the rate at which work is done.