Chapter 5Chapter 5Work and EnergyWork and Energy

Forms of EnergyForms of Energy

Mechanicalo Kinetic, gravitational

Thermalo Microscopic mechanical

Electromagnetic Nuclear

Energy is conserved!

WorkWork Relates force to change in energy

Scalar quantity Independent of time

cos)(

xFxxFW if

Units of Work and EnergyUnits of Work and Energy

xFW

SI unit = Joule1 J = 1 Nm = 1 kgm2/s2

Work can be positive or negativeWork can be positive or negative

Man does positive work lifting box

Man does negative work lowering box

Gravity does positive work when box lowers

Gravity does negative work when box is raised

Kinetic Kinetic EnergyEnergy

2

21mvKE Same units as work

Remember the Eq. of motion

xavv if 222

Multiply both sides by m/2,

xmamvmv if 22

21

21

xFKEKE if

Example 5.1Example 5.1A skater of mass 60 kg has an initial velocity of 12 m/s. He slides on ice where the frictional force is 36 N. How far will the skater slide before he stops?

120 m

Potential EnergyPotential Energy

If force depends on distance,

xFPE

For gravity (near Earth’s surface)

mghPE

Mysterious Rolling DemoMysterious Rolling Demo

Conservation of EnergyConservation of Energy

PEKEKEPEKEPE iiff

Conservative forces:• Gravity, electrical, QCD…Non-conservative forces:• Friction, air resistance…Non-conservative forces still conserve energy!Energy just transfers to thermal energy

Example 5.2Example 5.2

A diver of mass m drops from a board 10.0 m above the water surface, as in the Figure. Find his speed 5.00 m above the water surface. Neglect air resistance.

9.9 m/s

Example 5.3Example 5.3A skier slides down the frictionless slope as shown. What is the skier’s speed at the bottom?

H=40 m

L=250 m

start

finish

28.0 m/s

Two blocks, A and B (mA=50 kg and mB=100 kg), are connected by a string as shown. If the blocks begin

at rest, what will their speeds be after A has slid

a distance s = 0.25 m? Assume the pulley and incline are frictionless.

s

Example 5.4Example 5.4

1.51 m/s

Example 5.5Example 5.5

Three identical balls are thrown from the top of a building, all with the same initial speed. The first ball is thrown horizontally, the second at some angle above the horizontal, and the third at some angle below the horizontal as in the figure below. Neglecting air resistance, rank the speeds of the balls as they reach the ground.All 3 have same speed!

Springs (Hooke’s Springs (Hooke’s Law)Law)

kxF

Proportional to displacement from equilibrium

Potential Energy of SpringPotential Energy of Spring

PE=-Fx

x

F

xkxPE )(21

2

21kxPE

x

Example 5.6Example 5.6

b) Find the maximum distance d the block travels up the frictionless incline if the incline angle θ is 25°.

A 0.50-kg block rests on a horizontal, frictionless surface as in the figure; it is pressed against a light spring having a spring constant of k = 800 N/m, with an initial compression of 2.0 cm.

a) After the block is released, find the speed of the block at the bottom of the incline, position (B).

a) 0.8 m/s b) 7.7 cm

Example 5.7Example 5.7

A spring-loaded toy gun shoots a 20-g cork 40 m into the air after the spring is compressed by a distance of 1.5 cm.

a) What is the spring constant?

b) What is the maximum acceleration experienced by the cork?

a) 69,760 N/m b) 52,320 m/s2

Graphical connection between Graphical connection between FF and and PEPE

FxFPE

curveunderArea12 PEPE

xx1 x2x

Graphical connection between Graphical connection between FF and and PEPE

PE

xPE

F

xFPE

x

F = -slope, points down hill

Graphs of Graphs of FF and and PEPE for spring for spring

PE=(1/2)kx2

x

x

F=-kx

Force pushes you to bottom of potential well

Example 5.8Example 5.8

PE (

J)

x (m)

10

20

30

40

50

60

1.0 2.0 3.0 4.000

A particle of mass m = 0.5 kg is at a position x = 1.0 m and has a velocity of -10.0 m/s.What is the furthest points to the left and right it will reach as it oscillates back and forth?

0.125 and 3.75 m

Etot

PowerPower

Power is rate of energy transfer

tW

P

SI units are Watts (W)

3

2

s

mkg 1s/J 1W1

US Customary units are hp (horse power)

W746lb/sft 550hp 1

Example 5.9Example 5.9

An elevator of mass 550 kg and a counterweight of 700 kg lifts 23 drunken 80-kg students to the 7th floor of a dormitory 30 meters off the ground in 12 seconds. What is the power required? (in both W and hp)

41 kW =55.6 hp

Example 5.10Example 5.10

A 1967 Corvette has a weight of 3020 lbs. The 427 cu-in engine was rated at 435 hp at 5400 rpm.a) If the engine used all 435 hp at 100% efficiency during acceleration, what speed would the car attain after 6 seconds?b) What is the average acceleration? (in “g”s)

a) 119 mph b) 0.91g

Power: Force and velocityPower: Force and velocity

txF

tKE

P

FvP

For the same force, power is higher for higher v

Example 5.11Example 5.11

Consider the Corvette (w=3020 lbs) having constantacceleration of a=0.91ga) What is the power when v=10 mph?b) What is the power output when v=100 mph?

a) 73.1 hp b) 732 hp (in real world a is larger at low v)

Example 5.12Example 5.12A physics professor bicycles through air at a speed ofv=30 km/hr. The density of air is 1.29 kg/m3. The professor has cross section of 0.35 m2. Assume all of the air the professor comes in contact with is accelerated to v.

a) What is the mass of the air swept out by the professor in one second?

b) What is the power required to accelerate this air?a) 10.75 kg b) 373 W = 0.5 hp

Example 5.13Example 5.13Find the dimensions of the drag coefficient in the equation

3

2AvP is dimensionless

If =0.4, and the professor’s conversion of food to work is 20% efficient, how many kilocalories must the professor burn to cycle for 3 hours? (=1.29 kg/m3, A= 0.35 m2, v=30 km/hr, 1 kcal=4186 J)

673 kcalWhen we discuss diets cal.s actually refer to kcal!

Example 5.14Example 5.14

If a professor burns 75 W bicycling at 30 km/hr, how much power would burn bicycling at Lance Armstrong’s speed of 45 km/hr?

253 W

Ergometer DemoErgometer Demo

Example 5.15Example 5.15

A dam wishes to produce 50 MW of power. If the height of the dam is 75 m, what flow of water is required? (in m3/s)

68.9 m3/s = 1.80x104 gallons/s

Example 5.16Example 5.16

2001 cost of electricity

How much money does it cost to run a 100-W light bulb for one year if the cost of electricity is 8.0 cents/kWhr?

$ 70.08

Some energy factsSome energy factshttp://css.snre.umich.eduhttp://css.snre.umich.edu

US consumes 24% of Worlds energy (5% of population)

Each day, each of us consumes:o 3 gallons of oilo 20 lbs of coalo 221 cubic feet of natural gas

In 2000 the US consumed 9.9x1016 BTUs

1 BTU is energy required to raise 1 lb of H20 1 degree F1BTU = 1055 J

Einstein...Einstein...

2mcE c is velocity of light

For small velocities,22

21mvmcE

“Rest” energy

For any v,

2

22 1

c

vmcE

ExampleExample

Suppose one had a supply of anti-matter whichone could mix with matter to produce energy. What mass of antimatter would be required to satisfy the U.S. energy consumption in 2000? (9.9x1016 BTUs)

574 kg