Work and Energy

Work and Energy

Chapter 7

Physics chapter 7 2

Conservation of Energy

Energy is a quantity that can be converted from one form to another but cannot be created or destroyed.

The total amount of energy present in any process remains the same.

Physics chapter 7 3

Work

Has a very specific definition in physics.Involves a force moving a body through

a displacement.If the force is in the same direction as

the motion,

FsW

Physics chapter 7 4

Vector equation for work

If the force is directed at an angle to the displacement, then only the component of the force that is parallel to the displacement does work.

cosFsW

F

Physics chapter 7 5

Vector equation for work

Important – work is a scalar quantity, not a vector, even though it is derived from two vectors.

Work does not have a direction.

Physics chapter 7 6

Work

If q is between 0° and 90°, cos q is positive, so work done is positive.

If q is between 90° and 180°, cos q is negative, so work done is negative.

If q is 90°, cos q is zero, so work done is zero.

Physics chapter 7 7

Example

Lifting, carrying, and lowering a book.

Physics chapter 7 8

Units on work

The unit for work is a joule.

mN 1J 1

Physics chapter 7 9

Example

During your winter break you enter a dogsled race across a frozen lake. To get started you pull the sled (m = 80 kg) with a force of 180 N at 20° above the horizontal. You pull the sled 5 m.

Find the amount of work you do

Physics chapter 7 10

On your own

A truck of mass 3000 kg is to be loaded onto a ship by a crane that exerts an upward force of 31 kN on the truck. This force is applied over a distance of 2 m.

Find the work done by the crane on the truck


Work and energy with varying forces

The equations we’ve developed so far for work involve constant forces and motion along a straight line.

Now let’s look at varying forces.We could also look at motion along

curved lines, but we would need to use integrals, which you don’t know how to do yet.


The spring force

The farther you stretch a spring, the harder it is to stretch.

The force you must apply follows the following equation, known as Hooke’s Law

Where k is called the force constant of the spring or the spring constant. k has units of N/m

kxFx


Work done on a spring

2

2

1kXW

Where X is the total elongation of the spring.

See page 151.


Work done on a spring

21

22 2

1

2

1kxkxW

Where x1 is the initial position of the spring, x2 is the final position of the spring

x = 0 is the equilibrium position of the spring (when it is neither stretched nor compressed.)


Example

A 4-kg block on a frictionless table is attached to a horizontal spring with a force constant of 400 N/m. The spring is initially compressed with the block at x1 = -5 cm.

Find the work done on the block as the block moves to its equilibrium position x2 = 0


On your own

To stretch a spring 3.00 cm from its unstretched length, 12.0 J of work must be done. How much work must be done to compress this spring 4.00 cm from its unstretched length?


Work and Kinetic Energy

The total work done on a body is related to changes in the speed of the body.

asvv 221

22

s

vva

2

21

22


Work and Kinetic Energy

s

vvm

2

21

22

maF

2

21

22 vv

mFs

21

22 2

1

2

1mvmvFs


Work-Kinetic Energy Theorem

21

22 2

1

2

1mvmvWtot

The quantity ½ mv2 is called the kinetic energy and represented by the letter K. It is a scalar quantity and can never be negative. It is zero when an object is not moving.

KKKWtot 12


Units on kinetic energy

Looking at the equation, we have

2

2

s

mkg

ms

mkg

2 mN J


Work-kinetic energy theorem

The work-kinetic energy theorem still works for varying forces.

See the last paragraph on page 153 for a mathematical explanation of this.

21

22 2

1

2

1mvmvWtot


Motion along a curved path

Where dl is a distance along the path

The work-kinetic energy theorem still holds true.

2

1

P

P

dlFW


Example

During your winter break you enter a dogsled race across a frozen lake. To get started you pull the sled (m = 80 kg) with a force of 180 N at 20° above the horizontal. You pull the sled for 5 m.

FindThe work you doThe final speed of the sled


On your own

A truck of mass 3000 kg is to be loaded onto a ship by a crane that exerts an upward force of 31 kN on the truck. This force is applied over a distance of 2 m.

FindThe work done by the crane on the truckThe total work done on the truckThe upward speed of the truck after the 2 m

if it started from rest


Example

A 4-kg block on a frictionless table is attached to a horizontal spring with a force constant of 400 N/m. The spring is initially compressed with the block at x1 = -5 cm.

FindThe work done on the block as the block

moves to its equilibrium position x2 = 0The speed of the block at x2=0


Where does kinetic energy come from?

If an object falls off a cliff, gravity does work on it as it falls, and increases its kinetic energy.

But where does that energy come from?


Potential Energy

Energy seems to be stored in some form related to height.

This energy is related to the position of a body, not its motion.

It is called potential energy. – measures potential or possibility for work to be done.

(Some kinds of potential energy are related to things other than height.)


Work done by gravity

Work done by gravity as a body falls from height y1 to height y2.

0cosFsWgrav Fs 21 yyw

21 mgymgy

Also works if object is rising – then y2 is greater than y1, so the work is negative, as it should be.


Gravitational Potential Energy

We define the gravitational potential energy as

mgyU So, the work done by gravity is

21 UUWgrav 12 UU U


If only gravity does work

gravtot WW

UK

2112 UUKK

1122 UKUK


Conservation of Mechanical Energy

When only gravity does work, the total mechanical energy is constant.

constant UKE


Where do I measure y from?

Wherever you want.It is only the change in potential energy

that we are interested in, not the value of U at a particular point.

So choose your zero potential energy point wherever it is convenient.


Example

Standing near the edge of the roof of a 12-m high building, you kick a ball with an initial speed of vi = 16 m/s at an angle of 60° above the horizontal. Neglecting air resistance, use conservation of energy to find How high above the height of the building the ball

rises Its speed just before it hits the ground

9.79 m 22.2 m/s


You try

You throw a 0.200-kg ball straight up in the air, giving it an initial upward velocity of 20.0 m/s. Use conservation of energy to find how high it goes.

20.4 m


Effect of other forces

If forces other than gravity are acting on a body

othergravtot WWW 12 KK

1221 KKWUU other

2211 UKWUK other


Example

A child of mass 40 kg goes down an 8.0 m long slide inclined at 30° above the horizontal. The coefficient of kinetic friction between the slide and the child is 0.35. If the child starts from rest at the top of the slide, how fast is she traveling when she reaches the bottom?

5.60 m/s


Curved path

The shape of the path makes no difference for gravitational potential potential energy.

You can still use the same equations for conservation of energy.


On your own

A pendulum consists of a bob of mass m attached to a string of length L. The bob is pulled aside so that the string make an angle q0 with the vertical, and is released from rest. Find an expression for its speed as it passes through the bottom of the arc.

q0


Elastic potential energy

Think slingshot.The farther you stretch it, the more

kinetic energy the projectile can gain.


Springs

For springs, the elastic potential energy is given by

2

2

1kxU

The work done by a spring is

22

21 2

1

2

1kxkxWel

21 UU U

The equations work whether the spring is stretched or compressed.


Where is x = 0?

We don’t get to choose this timex = 0 is always at the equilibrium position

of the spring – when it is neither stretched nor compressed.


Conservation of energy

The conservation of energy works the same for springs as it does for gravity.

2211 UKWUK other


Conservation of energy

If we have both gravitational and elastic potential energies, then

2,2,21,1,1 elgravotherelgrav UUKWUUK


Example

A 1.20-kg piece of cheese is placed on a vertical spring of negligible mass and force constant k = 1800 N/m that is compressed 15.0 cm. When the spring is released, how high does the cheese rise from its initial position? (The spring and the cheese are not attached.)

1.72 m


On your own

A 2000-kg elevator with broken cables is falling at 25 m/s when it first contacts a cushioning spring at the bottom of the shaft. The spring is supposed to stop the elevator, compressing 3.00 m as it does so. During the motion, a safety clamp applies a constant 17,000 N frictional force to the elevator.

What is the force constant of the spring?

1.41 x 105 N/m


Conservative forces

The work done by conservative forces:Can always be expressed as the difference

between the initial and final values of a potential energy function.

Is reversible Is path-independent (It only depends on the

starting and ending points.)Equals zero when the starting and ending

points are the same.


Conservative forces

Gravitational (without air resistance)Spring (without friction)


Nonconservative forces

Depend on pathAre not reversibleCannot be expressed in terms of a

potential energy functionDo work even when the starting and

ending points are the same



FrictionChemical reaction forces



Increase or decrease the internal energy of a system

Frequently, this happens through heat.

otherWU int


Law of conservation of energy

This is always true – the most general form of the equation.

0int UUK


Power

Power is not the same as energy or force.

Power is the time rate at which work is done.

t

WPav


Power unit

The unit of power is the watt

s

J1 W1


The kWh

Your electric bill tells you how many kWh of electricity you used over the last month.

This is a unit of energy or work, not a unit of power

MJ 3.6or J 106.3h 1

s 3600

kW 1sJ

1000kWh 1 6


Power in terms of velocity and force

t

WPav

t

sFP parallelav

t

sFparallel

avparallelvF

vFP parallel cosFvP


Example

Force A does 5 J of work in 10 s.Force B does 3 J of work in 5 s.Which force delivers greater power?


On your own

A 5-kg box is being lifted upward at a constant velocity of 2 m/s by a force equal to the weight of the box.What is the power input of the force?How much work is done by the force in 4 s?

Work and Energy

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