Chapter 10 Chapter 10 Work and EnergyWork and Energy

Objectives: Objectives:

• Describe Describe work work in terms of force and in terms of force and displacement, using the definition of the displacement, using the definition of the scalar productscalar product..

• Solve problems involving concept of Solve problems involving concept of work.work.

• Distinguish between the Distinguish between the resultantresultant work work and the work of a single force.and the work of a single force.

• Define the Define the spring constantspring constant and calculate and calculate the work done by a varying spring force.the work done by a varying spring force.

Three things are necessary for the Three things are necessary for the performance of work:performance of work:

F

F

x

• There must be an applied force There must be an applied force F.F.

• There must be a displacement There must be a displacement x.x.

• The force must have a The force must have a component along the component along the displacement.displacement.

Definition of WorkDefinition of Work

WorkWork is anything that can exert a is anything that can exert a force through a distance.force through a distance.

•a a scalar quantityscalar quantity

WorkWork is anything that can exert a is anything that can exert a force through a distance.force through a distance.

•a a scalar quantityscalar quantity

Work = Force component X displacement

Work = Force component X displacement

1 N1 Nmm = = 1 Joule 1 Joule (J)(J)

Unit:

Work can Work can be:be:

Fx

• Positive, when force applies in the Positive, when force applies in the same direction as the same direction as the displacement.displacement.

• Negative, if the force is opposite Negative, if the force is opposite displacement directiondisplacement direction

If a force does not affect If a force does not affect displacement, it does no displacement, it does no

work.work.

F

W

The force The force F F exerted on exerted on the pot by the man does the pot by the man does work.work.The earth exerts a force The earth exerts a force W W on pot, but does no work on pot, but does no work even though there is even though there is displacement.displacement.

• Work done by any force that is at right angles with displacement will be zero (0).

Resultant work is the algebraic Resultant work is the algebraic sum of the individual works of sum of the individual works of

each force.each force.

Example:Example: F = F = 4040 N, f = -10 N and N, f = -10 N and x = x = 4 m4 m

Work = Work = (40 N)(4 m) + (-10 N)(4 m)(40 N)(4 m) + (-10 N)(4 m) Work = 120 JWork = 120 J

Resultant Work or Net Resultant Work or Net WorkWork

Fx f

OR

Example:Example: Work = (F - f) x

Work = (40 - 10 N)(4 m)

Example 1:Example 1: A lawn mower is pushed a A lawn mower is pushed a horizontal distance of horizontal distance of 20 m20 m by a force of by a force of 200 N200 N directed at an angle of directed at an angle of 303000 with the with the ground. What is the work of this force?ground. What is the work of this force?

300

x = 20 m

F = 200 N

Work = Work = (F cos (F cos ) x) xWork = Work = (200 N)(20 m) Cos (200 N)(20 m) Cos

303000

Work = 3460 JWork = 3460 J

Note: Work is positive since Fx and x are in the same direction.

FF

Graph of Force vs. Graph of Force vs. DisplacementDisplacement

Assume that a constant force F acts Assume that a constant force F acts through a parallel displacement through a parallel displacement x.x.

Force, FForce, F

Displacement, xDisplacement, x

FF

xx11 xx22

The The area area under the under the curve is equal to the curve is equal to the work done.work done.

Work = F(xWork = F(x22 - x - x11))

Work F x

Area

Example for Constant ForceExample for Constant ForceWhat work is done by a constant force of What work is done by a constant force of 40 N 40 N moving a block from x = moving a block from x = 1 m1 m to x = to x =

4 m4 m??

Work = F(xWork = F(x22 - x - x11))

Work F x Work F x 40 N40 N

Force, FForce, F

Displacement, xDisplacement, x

1 m1 m 4 4 mm

AreaWork = Work = (40 N)(4 m - 1 (40 N)(4 m - 1

m)m)

Work = 120 JWork = 120 J

Work of a Varying ForceWork of a Varying Force

Our definition of work applies only for Our definition of work applies only for a a constantconstant force or an force or an averageaverage force. force.

What if the force What if the force variesvaries with with displacement as with stretching a displacement as with stretching a

spring or rubber band?spring or rubber band?

FF

xx FF

xx

Hooke’s LawHooke’s LawWhen a spring is stretched, there is a When a spring is stretched, there is a

restoringrestoring force that is proportional to the force that is proportional to the displacement.displacement.

F = -kxF = -kx

The spring constant The spring constant kk is a is a property of the spring given property of the spring given by:by:

K = F

x

FF

xx

m

Work Done in Stretching a Work Done in Stretching a SpringSpring

F

x

m

Work done Work done ONON the spring is the spring is positivepositive; work ; work BYBY the spring is the spring is

negative.negative.

From Hooke’s law: F = kxFrom Hooke’s law: F = kx

x

F

Work = Area of Triangle Work = Area of Triangle Area = Area = ½½ (base)(height) (base)(height)

= = ½ ½ (x)(F(x)(Favgavg ) = ) = ½ ½ x(kx)x(kx)

Work = ½ kx2Work = ½ kx2

Compressing or Stretching a Compressing or Stretching a Spring Initially at Rest:Spring Initially at Rest:Two forces are Two forces are always present: always present: the outside force the outside force FFoutout ON ON spring spring and the reaction and the reaction force force FFss BY BY the the spring.spring.Compression: Fout does positive work and

Fs does negative work (see figure).Stretching: Fout does positive work and Fs does negative work (see figure).

x

m

x

mCompressin

g Stretching

General Case for Springs:General Case for Springs:

If the initial displacement is not zero, If the initial displacement is not zero, the work done is given by:the work done is given by:

2 21 12 12 2Work kx kx 2 21 1

2 12 2Work kx kx

x1 x2

Fx1

m

x2

m

PowerPowerPower is defined as the rate at

which work is done: (P = dW/dt )

Power of 1 W is work done at rate of 1 J/s

Power of 1 W is work done at rate of 1 J/s

Work F rPower

time t

Work F rPower

time t

Ft

SF

t

xFPower

Units of PowerUnits of Power

1 W = 1 J/s and 1 kW = 1000 W

One watt (W) is work done at the rate of one joule per second.

One ft lb/s is an older (USCS) unit of power.

One horsepower is work done at the rate of 550 ft lb/s. ( 1 hp = 550 ft lb/s

= 746 W)

Example 4:Example 4: A 100-kg cheetah moves A 100-kg cheetah moves from rest to 30 m/s in 4 s. What is the from rest to 30 m/s in 4 s. What is the power?power?

Recognize that work is equal Recognize that work is equal to the change in kinetic to the change in kinetic

energy:energy: WorkP

t2 21 1

02 2fWork mv mv

2 21 12 2 (100 kg)(30 m/s)

4 sfmv

Pt

m = 100 kg

Power Consumed: P = 1.22 kW

Power Consumed: P = 1.22 kW

CONCLUSION:CONCLUSION:Chapter 10 - WorkChapter 10 - Work