Energy and Work

Energy - the ability of a body or system of bodies to perform work.

A body is given energy when a force does work on it.

But What is Work?

A force does work on a body (and changes its energy) when it causes a displacement.

If a force causes no displacement, it does no work.

Riddle Me This

If a man holds a 50 kg box at arms length for 2 hours as he stands still, how

much work does he do on the box?

Nad

a ZipZilch

NONEZERO

Counterintuitive Results There is no work done by a force

if it causes no displacement. Forces perpendicular to

displacement, such as the normal force, can do no work.

Likewise, centripetal forces never do work.

Calculating WorkWork is the dot product of

force and displacement.

Work is a scalar resulting from the interaction of two vectors.

Vector Multiplication

There are three ways to multiply vectors:

•Scalar Multiplication

•Dot Product

•Cross Product

Scalar Multiplication

•Magnitude of vector changes.•Direction of vector does not change.

amF

a = 10 m·s-1

F = 50 N

If m = 5 kg

Dot Product

BAW

Note that the dot

product of two vectors gives a scalar .

. and between angle theis BA

cosABBA

θ

A

B

Dot Product

Geometrically, the dot product is the projection of one vector on a second

vector multiplied by the magnitude of the second vector.

θ

A

B

cosA

Calculating WorkcosFssFW

dxxFW )(

θ

F

s

cosF

Which does more work?

θ

F1

F2

Two forces are acting on the box shown causing it to move across the floor. Which force does more work?

Vectors and Work

F

Vectors and Work

Fs

W = F • sW = F s cos 0o

W = F sMaximum positive work

Vectors and Work

F

Vectors and Work

s

W = F • sW = F s cos Only the component of force aligned with displacement does work. Work is less.

F

Vectors and Work

F

Vectors and Work

F s

W = F • sW = F s cos 180o

W = - F sMaximum negative work.

Gravity often does negative work.

mg

F

When the load goes up, gravity does negative work and the crane does positive work.

When the load goes down, gravity does positive work and the crane does negative work.

Positive, zero, or negative work?

A box is being moved with a velocity v by a force P (parallel to v) along a level floor. The normal force is FN, the frictional force is fk, and the weight of the box is mg.

Decide which forces do positive, zero, or negative work.

Positive, zero, or negative work?

v

mg

P

FN

fk

s

Units of Work

cosFssFW

J = N·m

J = kg·m2·s-2

Work and variable force

The area under the curve of a graph of force vs displacement gives the work done by the force.

F(x)

xxa xb

W = F(x) dxxa

xb

Net Work

Net work (Wnet) is the sum of the work done on an object by all forces acting

upon the object.

The Work-Energy Theorem

• Consider a force applied to an object (ΣF ≠ 0).

• Newton’s second law tells us that this net force will produce an acceleration.

• Since the object is accelerating, its displacement will change, hence the net force does work.

WsF as

vv if 2

)( 22

2212

21

if mvmvW

The Work-Energy Theorem

maF massF masW

asvv if 222

2

)( 22if vv

mW

Kinetic Energy

A form of mechanical energyEnergy due to motionK = ½ m v2

– K: Kinetic Energy in Joules.

– m: mass in kg

– v: speed in m/s

2212

21

if mvmvW

The Work-Energy Theorem

maF masW

KEWnet

The Work-Energy Theorem

Wnet = KE– When net work due to all forces acting upon

an object is positive, the kinetic energy of the object will increase.

– When net work due to all forces acting upon an object is negative, the kinetic energy of the object will decrease.

– When there is no net work acting upon an object, the kinetic energy of the object will be unchanged.

PowerPower is the rate of which work is

done.No matter how fast we get up the

stairs, our work is the same.When we run upstairs, power demands

on our body are high.When we walk upstairs, power

demands on our body are lower.

PowerThe rate at which work is

done.Pave = W / t

P = dW/dtP = F • v

Units of Power

Watt = J/sft lb / shorsepower

•550 ft lb / s

•746 Watts

Power Problem

Develop an expression for the power output of an airplane cruising at constant speed v in level flight. Assume that the aerodynamic drag force is given by FD = bv2. By what factor must the power be increased to increase airspeed by 25%?

How We Buy Energy…The kilowatt-hour is a commonly used unit

by the electrical power company.Power companies charge you by the

kilowatt-hour (kWh), but this not power, it is really energy consumed.

1 kW = 1000 W1 h = 3600 s

1 kWh = 1000J/s • 3600s = 3.6 x 106J

More about force types

Conservative forces:– Work in moving an object is path independent.– Work in moving an object along a closed path is zero.– Work may be related to a change in potential energy

or used in the work-energy theorem.– Ex: gravity, electrostatic, magnetostatic, springs

Non-conservative forces:– Work is path dependent.– Work along a closed path is NOT zero.– Work may be related to a change in total energy

(including thermal energy).– Ex: friction, drag, magnetodynamic

Mechanical Energy:Potential energyEnergy an object possesses by virtue

of its position or configuration.Represented by the letter U.Examples:

– Gravitational potential energy.– Electrical potential energy.– Spring potential energy.

Gravitational Potential Energy (Ug)

For objects near the earth’s surface, the gravitational pull of the earth is constant, so

Wg = mgx – The force necessary to lift an object at

constant velocity is equal to the weight, so we can say

Ug = -Wg = mgh

Where is Gravitational Potential Energy Zero?Ug has been defined to be zero when

objects are infinitely far away, and becoming negative as objects get closer.

This literal definition is impractical in most problems.

It is customary to assign a point at which Ug is zero. Usually this is the lowest point an object can reach in a given situation

Then, anything above this point is a positive Ug.

Ideal SpringObeys Hooke’s Law.

Fs(x) = -kx

– Fs is restoring force exerted BY the spring.

Ws = Fs(x)dx = -k xdx

– Ws is the work done BY the spring.

Us = ½ k x2

Spring ProblemThree identical springs (X, Y, and Z) are hung as shown. When a 5.0-kg mass is hung on X, the mass descends 4.0 cm from its initial point. When a 7.0-kg mass is hung on Z, how far does the mass descend?

X Y

Z

System

Boundary

Isolated System

E = U + K + Eint

= Constant

No mass can enter or leave!No energy can enter or leave!Energy is constant, or conserved!

Boundary allows no exchange with environment.

Law of Conservation of Energy

Energy can neither be created nor destroyed, but can only be transformed from one type of energy to another.

Applies to isolated systems.

Law of Conservation of Mechanical Energy

E = U + K = CE = U + K = 0

for gravityUg = mghf - mghi

K = ½ mvf2 - ½ mvi

2

Law of Conservation of Mechanical Energy

E = U + K = CE = U + K = 0

for springsUg = ½ kx2

K = ½ mvf2 - ½ mvi

2

Law of Conservation of Energy

E = U + K + Eint= CE = U + K + Eint = 0

Eint is thermal energy.Mechanical energy may be converted to and from heat.

James Prescott Joule

Father of Conservation of Energy.Studied electrical motors.Derived the mechanical equivalent of heat.Measured heat of water as it fell.Measured cooling of expanding gases.

h

Pendulum Energy

½mvmax2 =

mghFor minimum and maximum points of swing

½mv12 + mgh1 = ½mv2

2 + mgh2 For any points two points in the pendulum’s swing

Spring Energy

m

m -x

mx

0½ kx2 = ½ mvmax

2

For maximum and minimum displacements from equilibrium½ kx1

2 + ½ mv12

= ½ kx22 + ½

mv22

For any two points in a spring’s oscillation

Announcements 04/21/23

Clicker Quiz.

Homework Check.

Spring and Pendulum Energy Profile

E

x

Total Energy

UK

EquilibriumThe net force on a system is zero

when the system is at equilibrium.

There are three types of equilibrium which describe what happens to the forces on a system when it is displaced slightly from the equilibrium position.

Force and Potential Energy

In order to discuss the relationships between displacements and forces, we need to know a couple of equations.

W = F(x)dx = - dU = -U

dU = - F(x)dx

F(x) = -dU(x)/dx

Stable Equilibrium

U

x

compressed extended

dU/dx

F = -dU/dx

When x is negative, dU/dx is negative, so F is positive and pushes system back to equilibrium.

Stable EquilibriumExamples:

– A spring at equilibrium position.

When the system is displaced from equilibrium, forces return it to the equilibrium position.

Often referred to as a potential energy well or valley.

Stable Equilibrium

U

x

compressed extended

dU/dxWhen x is positive, dU/dx is positive, so F is negative and pushes system back to equilibrium.

F = -dU/dx

Stable Equilibrium

U

x

compressed extended

dU/dxWhen x is zero, dU/dx is zero, so F is zero and there are no forces on the system pushing it anywhere.

F = -dU/dx

Unstable EquilibriumExamples:

– A cone on its tip.

When the system is displaced from equilibrium, forces push it farther from equilibrium position.

A potential energy peak or mountain.

Unstable Equilibrium

U

x

dU/dx

F = -dU/dx

When x is negative, dU/dx is positive, so F is negative and pushes system further from equilibrium.

Unstable Equilibrium

U

x

dU/dxF = -dU/dx

When x is positive, dU/dx is negative, so F is positive and pushes system further from equilibrium.

Unstable Equilibrium

U

x

dU/dxF = -dU/dx

When x is zero, dU/dx is zero, so F is zero and no forces are trying to push the system anywhere.

Neutral EquilibriumExamples:

– A book on a desk.

When the system is displaced from equilibrium, it just stays there.

A potential energy plane.

Neutral Equilibrium

U

x

F = -dU/dx

When x changes, dU/dx is zero, so F is zero and no force develops to push the system toward or away from equilibrium.

dU/dx

Potential Energy and Force•F(r) = -dU(r)/dr

•Fx = -U/ x•Fy = -U/ y•Fz = -U/ z

Equilibrium equationsStable

U/x = 0, 2U/x2 > 0Unstable

U/x = 0, 2U/x2 < 0Neutral

U/x = 0, 2U/x2 = 0

Atomic bonds and Equilibrium

x

U

Lowest energy inter-atomic separation

Atoms too close

Atoms too far apart