Work, Energy, and Power

“It is important to realize that in physics today, we have no knowledge of what energy is.” - R.P. Feynman

Work and EnergyWork and Energy concepts (and also momentum, later) provide an alternative, easier approach to mechanics!

• Measures a change in the condition of matter (change can be in velocity, position, mass, etc.)

ENERGY

• There is no such thing as pure energy. Energy is the measure of a change (and force is the agent of the change).

• Causes transfer of energy (between masses) or transforms energy (from one type to another).

WORK

• work is the amount of energy transferred by forces.

car gains kinetic energy

car loses kinetic energy

car maintains kinetic energy

F and dare parallel

F and dare perpendicular

F and dare opposite

Energy Example: Driving a Car

SPEED UP SLOW DOWN ‘ROUND A CORNER

F

d

F

d

Fd

Fa d

Fg

d

rocket gains potential energy

rocket gains kinetic energy

rocket has constant energy (circular orbit)

F and dare parallel

F and dare perpendicular

F and dare parallel

Fa

d

Energy Example: The Launch of a Rocket

LIFTOFF! SPEED! ORBIT!

Work

• force on an object

• displacement of an object

Work depends on three things:

• angle between the force and displacement (force must cause the displacement)

W =Fdcosθ

• work is a scalar quantity, but can be positive, negative, or zero because it represents the amount of energy change.

click for appletUnits (metric or SI)

1 joule =1 newton×1 meter=1 N⋅m Fa

PHYSICSFk

Fn

Fg dwork is positive when 0˚≤ θ < 90˚

work is negative when 90˚< θ ≤ 180˚

work is zero when θ = 90˚Which force does positive work?

Which does negative work?Which does zero work?

click for applet

(translational) Kinetic Energy - depends on the motion of macroscopic objects (e.g. a car in motion) moving linearly

Kinetic energy is the energy of motion of matter

Kinetic Energy

Thermal Energy - depends on the motion of microscopic objects (e.g. atomic vibrations). Technically not the same as heat.

KE = 12 mv

2

TE =Fkd

mass velocityin joules

friction distance

in joulesclick for applet

Wnet =ΔKEWnet =ΣFd ΣF =ma

Wnet =mad vf2 =vi

2 + 2ad

Wnet =m12 (vf

2 −vi2 )

work, dynamics, kinematics! WORK-ENERGY THEOREM

Potential Energy

Gravitational Potential - depends on the position of mass in a gravitational field

Potential energy is the energy of position of matter

Elastic Potential - depends on the position of mass on an atomic scale

GPE =mghg field

mass height

in joules

EPE = 12 kx

2

spring constant

position

in joules

Power

Power is the rate at which work is done (or energy is used)

Power can also be expressed in terms of force and velocity

P =Wt

Units (metric or SI)

1 watt =1 joule1 second

=1 Js

P =Wt=Fdcosθ

t=Fv

click for web page

Conservation of Mechanical Energy

click for animation

Conservative forces (gravity, spring force) keep mechanical energy constant.

KEi + PEi =KE f + PE f

Mechanical energy is the sum of kinetic and potential energy.

Potential and kinetic energy may change, but the total mechanical energy does not change.

ME =KE + PE

click for applet

Conservation of Mechanical EnergyExampleA spring with constant 800 N/m is compressed 10 cm. It is released against a cart with mass of 0.25 kg that moves along a track without friction.What is the cart’s speed when it leaves the spring?

What is the cart’s speed when it reaches the top of a 0.75 m high hill?

At what height above the original release point does the cart come to rest?

EPEi =KE f ⇒ 12 kxi

2 = 12 mvf

2

12 (800)(0.10)

2 = 12 (0.25)vf

2

vf =5.66 m/s

KEi =KE f +GPE f

12 mvi

2 = 12 mvf

2 +mgh12 (0.25)(5.66)

2 = 12 (0.25)vf

2 + (0.25)(9.8)(0.75)

vf =4.16 m/s

or 12 kxi2 = 1

2 mvf2 +mgh

KEi =GPE f

12 mvi

2 =mghf12 (0.25)(5.66)

2 =(0.25)(9.8)hfhf =1.63 m

or 12 mvi2 +mghi =mghf where hi =0.75 m

vi =4.16 m/s

or 12 kxi2 =mghf

click for animation

click for animationW + KEi + PEi =KE f + PE f +TE

Non-conservative forces (friction, applied, normal, tension) change the mechanical energy, but total energy remains constant.

Law of Conservation of Energy

result of friction

result of applied or normal or tension

result of gravity or spring force

Law of Conservation of EnergyExamples (Assume there is zero air friction in these problems)

A 75-kg Olympic ski jumper starts from rest and glides down a 30˚ incline 100-meter long. The track has surface friction. If the jumper leaves the track with a velocity of 28 m/s, what is the average force of kinetic friction on the skies from the track?

h =dsinθ =100sin30˚=50mGPEi =KE f +TE

mghi =12 mvf

2 + Fkd

(75)(9.8)(50) = 12 (75)(28

2 ) + Fk(100)Fk =73.5 N

In frustration, a physics student shoves a 1.2-kg textbook with a force of 14 newtons across a 0.5 meter wide desk that has no surface friction. If the book lands on the ground with a velocity of 5.2 m/s, how high is the desk above the ground?

W +GPEi =KE f

Fdcosθ +mghi =12 mvf

2

(14)(0.5)cos0˚+(1.2)(9.8)(hi ) =12 (1.2)(5.2

2 )hi =0.784 m

Honors: A 400-g wood block is attached to a spring. The block can slide along a table with coefficient of friction 0.25. A force of 20 N compresses the spring 20 cm and the block is released. How far beyond the equilibrium position will the spring stretch?

k =Fsp / xi =20 / 0.2 =100 N/m

EPEi =EPE f +TE12 kxi

2 = 12 kxf

2 + Fk(xi + xf )

Fk =μkmg=(0.25)(0.4)(9.8) =0.98 N12 (100)(0.2

2 ) = 12 (100)(xf

2 ) + (0.98)(0.2 + xf )

xf =0.18 m