Chapter 6Work and Energy

1

0) Energy & conservation laws• Newton’s laws (with kinematic equations) allow a

complete and continuous description of motion, but requires

– knowledge of the force at all times – detailed calculations which may be prohibitive

• Conservation laws allow intermediate details to be ignored by realizing some quantities are conserved

2

Conservation of momentum: p = mv - discovered first (consequence of Newton’s laws) - mv was considered an essential quality of motion

Descartes

3

What kind of essential motion remains zero when bodies are being hurtled through space? Found mv2 conserved in collisions.

4

1) Energy

1807: Associated “Energy” with mv2

5

It is important to realize that in physics today, we have no knowledge of what energy is

6

As we cannot give a general definition of energy, the principle of the conservation of energy simply signifies that there is something which remains constant. Well, whatever new notions of the world future experiments may give us, we know beforehand that there will be something which remains constant and which we will be able to call energy. Henri Poincaré (1854 - 1912)

7

• Energy is some quantity, which takes many forms, associated with a system which we know intuitively is conserved. Or else we could make a perpetual motion machine. Feynman writes that the conservation of energy is the statement that perpetual motion is not possible.

• State of a system in relation to fundamental forces • Force is the agent of change; energy is measure of

change

8

a) Work along a straight line: F || s

2) WorkWork produces a change in the energy of a system by the application of a force acting over a distance.

F

m

s

W = FsUnits: 1 N m = 1 joule = 1 J

Work done by force F, is

If the displacement s is zero, no work is done.

9

(b) Force and displacement not parallel

F

m

s

θ

The change in the system is the same as if a parallel force Fs = F cos θ acted over a distance s, so the work done by F is

W = Fss = Fs cosθ

10

(c) Negative work If a force acts opposite to the direction of

motion (like friction), it does negative work. It takes energy out of the system.

fm

s

v v = 0θ

W = Fs cosθ = Fs cos180º = −Fs

11

Example: A person pulls a block (mass 10 kg) up an inclined plane at constant speed. The block moves 1 m along the plane and the tension force acting on the block is parallel to the plane. The coefficient of friction is μk = 0.20. Find the work done on the block by (a) the tension force (b) gravity (c) the normal force (d) the kinetic friction force (e) the net force

�

Wnet =Wg +WT +Wf +WN

12

A cable lifts a 1200-kg elevator at a constant velocity for a distance of 35 m. What is the work done by (a)  the tension in the cable and (b) the elevator's weight?

13

3) Kinetic Energy• Work produces a change in energy. • Work on an otherwise free object produces

motion. • The energy associated with motion is called

kinetic energy, KE. • Work - energy theorem: If as a result of work on an

object (system) the only change is its resulting motion:

W = ΔKE = KEf − KEi

14

F

m

s

vv0

2nd law: F = makinematics: v2 = v0

2 + 2as

Using

Fs = mas = m (v2 − v0

2 )2

= 12mv2

gives for an object starting from rest:

Define Kinetic Energy:

KE = 12 mv

2

so that

W = ΔKE Work ∆KE=

ΔKE = 12 mv

2 − 12 mv0

2

15

• Work-energy theorem qualifications: – Force is the net force on the object (system) – No internal changes to object (system) e. g. Lifting a rock produces no kinetic energy because there is

no net force on the rock (on average). e.g. Compressing a spring produces no kinetic energy because

there are internal changes to the system

W = ΔKE

16

A 3.00-kg model airplane has velocity components of 5.00 m/s due east and 8.00 m/s due north. What is the plane’s kinetic energy?

a) 134 J b) 96 J c) 38 J d) 254 J

17

A 0.075-kg arrow is fired horizontally. The bowstring exerts an average force of 65 N on the arrow over a distance of 0.90 m. With what speed does the arrow leave the bow?

18

C&J 6.21An asteroid is moving along a straight line. A force acts along the displacement of the asteroid and slows it down. The asteroid has a mass of 4.5 x 104 kg, and the force causes its speed to change from 7100 to 5500 m/s. (a)  What is the work done by the force? (b)  If the asteroid slows down over a distance of 1.8E6 m, determine the magnitude of the average force.

19

4) Potential Energya) Work on a system

F = mg

m

h

• Lifting an object does not produce kinetic energy.

• Rather, it changes the configuration of the earth/object system.

• If the object is released, then it gains kinetic energy.

• The system stores energy by virtue of its configuration

20

• Potential Energy is energy stored in the configuration of interacting objects.

21

• Work done by a conservative force is reversible • Work done by a conservative force is independent of path • Work done by a conservative force in a closed loop is zero

– e.g. gravity, spring, electromagnetic • all fundamental forces • any force determined by the configuration of the system

W = −mgh

m

mg

m

mg

W = mgh

m

mg

W = mg cos 90º = 0

e.g. Work done by gravity

m

mg

W = −mgh + 0 +mgh + 0 = 0

(b) Conservative force

22

• Dissipative forces like friction and air resistance are non-conservative. The work depends on the path.

• No potential energy can be associated with such forces.

23

C&J conceptual question 12 Air resistance is a nonconservative force. It always opposes the motion of an object. An airplane flies from New York to Atlanta and then returns to its point of departure. The net work done by air resistance during this round trip ___________.

a) is zero b) is positive c) is negative d) is negative for slow speeds and positive for high speeds. e) is positive for slow speeds and negative for high speeds.

24

• If, as a result of work (Wext) on a system (against conservative forces), the only change is its configuration:

Wext (against cons forces) = ΔPE = PEf - PEi

Work done by conservative forces in a system produces a negative change in the potential energy.

Wc (by cons forces) = −ΔPE

25

5) Gravitational potential energy (near earth) Object lifted by an external force:

F = mg

m

h

ΔPE =Wext = mgh

The change in the potential energy is the work done (if there are no other changes in the system):

The position of zero PE is arbitrary; only changes are interesting. Define PE = 0 for h = 0. Then,

PE = mghWork done by gravity is -mgh, so Wg = −ΔPE

26

ΔPE = mghf −mgh0

Wg = −ΔPE

Wc = −ΔPE

(quantitative definition)

27

7) Conservation of Mechanical EnergyMechanical energy: E = PE + KE

If all forces in an isolated system are conservative, mechanical energy is constant

Work-energy: W = ΔKE

If all work is done by conservative forces: W = -ΔPE

Therefore, -ΔPE = ΔKE, or

E = PE + KE = constant

28

8) External (or non-conservative) forcesWork-energy: W = ΔKE

Consider work done by conservative and non-conservative (or external) forces: W = Wc + Wnc

Then

Wnc (or Wext ) = ΔPE + ΔKE

Work done by conservative forces can be accounted for by the change in potential energy: Wc = -ΔPE

work not accounted for by potential energy changes

29

Wnc (or Wext ) = ΔPE + ΔKE

Δ(mgh) Δ( 12 mv2 )

Fscosθ for forces not accounted for by PE

Summary

30

October 2005 midterm exam Q 6 A box bounces off a truck while it is at rest on a bridge and falls into a river that is 160 m below. As it falls, 40% of its energy is lost due to air resistance. What is its speed at the moment it strikes the water?

31

Example 8 A Daredevil Motorcyclist !A motorcyclist is trying to leap across the canyon by driving horizontally off a cliff 38.0 m/s. Ignoring air resistance, find the speed with which the cycle strikes the ground on the other side.

32

A person slides down from rest a large frictionless spherical surface. At what angle θ does the person leave the surface? (When the person leaves the surface, the normal force is zero)

33

6.84 A 63-kg skier coasts up a snow-covered hill that makes an angle of 25° with the horizontal. The initial speed of the skier is 6.6 m/s. After coasting 1.9 m up the slope, the skier has a speed of 4.4 m/s. (a)  Find the work done by the kinetic frictional force that acts on the skis. (b)  What is the magnitude of the kinetic frictional force?

34

9) Power• Time rate of change of work or energy • Average power:

P = Wnet

Δt= ΔE

Δt

Units: 1 watt = 1 W = 1 J/s

(1 hp = 745.7 W = ~ 3/4 kW)

35

C&J 6.67The cheetah is one of the fastest-accelerating animals, because it can go from rest to 27 m/s (about 60 mi/h) in 4.0 s. If its mass is 110 kg, determine the average power developed by the cheetah during the acceleration phase of its motion. Express your answer in watts.

36

• Power and speed

P = WΔt

= FsΔt

for constant force parallel to motion

but v = sΔt

so P = Fv

37

Oct 2005 midterm exam. Q8 A 50 kg girl runs up a flight of stairs in a time of 12.5 s. The stairs are 7.0 m long and make an angle of 27° above the horizontal. What is the average power that she produced?

38

10) Variable force• For constant force parallel to motion

W = Fs• Represents area of rectangle F by s

s

F

Force

displacement39

Force

Approximate variable force by a series of constant forces

Force

F1

F2

Total work is sum of rectangular areas, which approximates the area under the curve. As steps get smaller, the approximations approach equality:

W = area under F vs s curve

40

41

A basketball of mass 0.60 kg is dropped from rest from a height of 1.05 m. It rebounds to a height of 0.57 m. (a)  How much mechanical energy was lost during the collision with the floor? (b)  A basketball player dribbles the ball from a height of 1.05 m by exerting a constant downward force on it for a distance of 0.080 m. In dribbling, the player compensates for the mechanical energy lost during each bounce. If the ball now returns to a height of 1.05 m, what is the magnitude of the force?

42

11) Conservation of Energy• Energy can be neither created nor destroyed, but

can be converted from one form to another: – mechanical – heat – chemical – electrical – nuclear

43