WorkThe VERTICAL component of the force DOES

NOT cause the block to move the right. The energy

imparted to the box is evident by its motion to the

right. Therefore ONLY the HORIZONTAL

COMPONENT of the force actually creates energy

or WORK.

When the FORCE and DISPLACEMENT are in

the SAME DIRECTION you get a POSITIVE

WORK VALUE. The ANGLE between the force

and displacement is ZERO degrees.

When the FORCE and DISPLACEMENT are in

the OPPOSITE direction, yet still on the same axis,

you get a NEGATIVE WORK VALUE. IT simply

means that the force and displacement oppose each

other. The ANGLE between the force and

displacement in this case is 180 degrees.

When the FORCE and DISPLACEMENT are

PERPENDICULAR, you get NO WORK!!! The

ANGLE between the force and displacement in this

case is 90 degrees.

Scalar Dot Product?

cosxFxFW

A dot product is basically a CONSTRAINT

on the formula. In this case it means that

F and x MUST be parallel. To ensure that

they are parallel we add the cosine on the

end.

FORCE

Displacement

xFW

xFxFW

10cos;0

cos

cosxFxFW

FORCE

Displacement

xFW

xFxFW

f

1180cos;180

cos

cosxFxFW

FORCE

Displacement

JW

xFxFW

0

090cos;90

cos

Work = The Scalar Dot Product between

Force and Displacement. So that means if

you apply a force on an object and it covers

a displacement you have supplied ENERGY

or done WORK on that object.

nt vectordisplaceme

cos

r

rFrFW

Example

A box of mass m = 2.0 kg is moving

over a frictional floor has a force

whose

magnitude is F = 25 N applied to it at

an angle of 30 degrees. The box is

observed to move 16 meters in the

horizontal direction before falling off

the table.

a) How much work does F do before

taking the plunge?

cosrFW

rFW

JW

NmW

W

rFW

rFW

4.346

4.346

30cos1625

cos

Example

What if we had done this in UNIT VECTOR

notation?

JW

NmW

W

rFrFW

jiF

yyxx

4.346

4.346

)05.12()1665.21(

)()(

ˆ5.12ˆ65.21

Work-Energy Theorem

22

)22

()|2

|(

)(

22

222

o

ov

v

v

v

mvmvW

vvm

vmW

dvvmW

dvvmdvdt

dxmW

o

o

KW

mvK

2

21Energy Kinetic

Kinetic energy is the ENERGY of MOTION.

Example

Suppose the woman in the figure above applies a 50 N force

to a 25-kg box at an angle of 30 degrees above the

horizontal. She manages to pull the box 5 meters.

a) Calculate the WORK done by the woman on the box

b) The speed of the box after 5 meters if the box started from

rest.

30cos)5)(50(

cos

W

xFW

v

vW

mvKEW

2

2

)25(2

1

21

216.5 J 4.16 m/s

Conservative Force

Conservative Forces

◦ A force where the work it does is independent of

the path

x

y

z

A

BF

A conservative force depends only on the position of the

particle, and is independent of its velocity or acceleration.

Something is missing….

UW

ymgW

yymgW

FrrFW

gravity

gravity

gravity

gravity

180cos)(

cos

21

UW

ymgW

yymgW

FrrFW

gravity

gravity

gravity

gravity

0cos)(

cos

12

Consider a mass m that moves from position 1 ( y1)

to position 2 m,(y2), moving with a constant velocity.

How much work does gravity do on the body as it

executes the motion?

Suppose the mass was thrown UPWARD.

How much work does gravity do on the

body as it executes the motion?

In both cases, the negative

sign is supplied

The bottom line..

The amount of Work gravity does on a

body is PATH INDEPENDANT. Force

fields that act this way are

CONSERVATIVE FORCES

FIELDS. If the above is true, the amount

of work done on a body that moves

around a CLOSED PATH in the field will

always be ZERO

FRICTION is a non conservative force. By NON-

CONSERVATIVE we mean it DEPENDS on the PATH. If a

body slides up, and then back down an incline the total

work done by friction is NOT ZERO. When the direction of

motion reverses, so does the force and friction will do

NEGATIVE WORK in BOTH directions.

Nonconservative Forces

Examples of Nonconservative forces

◦ Friction

◦ Air resistance

◦ Tension

◦ Normal force

◦ Propulsion force of things like rocket

engine

Each of these forces depends on the

path

Potential Energy

mg

h

PEmghW

hxmgFxFW

10cos,0

;cos

Since this man is lifting the

package upward at a CONSTANT

SPEED, the kinetic energy is NOT

CHANGING. Therefore the work

that he does goes into what is

called the ENERGY OF POSITION

or POTENTIAL ENERGY.

All potential energy is considering

to be energy that is STORED!

Conservation of Energy

A B C D

In Figure A, a pendulum is

released from rest at some

height above the ground

position.

It has only potential energy.

In Figure C, a pendulum is at

the ground position and

moving with a maximum

velocity.

It has only kinetic energy.

In Figure D, the pendulum has

reached the same height

above the ground position as

A.

It has only potential energy.

In Figure B, a pendulum is still

above the ground position, yet

it is also moving.

It has BOTH potential energy

and kinetic energy.

Hooke’s Law

If a ‘spring’ is bent, stretched or compressed from its equilibrium position, then it will exert a restoring force proportional to the amount it is bent, stretched or compressed.

Fs = kx (Hooke’s Law)

Related to Hooke’s Law, the work required to bend a spring is also a function of k and x.

Ws = (½ k)x2 = amount of stored strain energy◦ K is the stiffness of the spring. Bigger means Stiffer

◦ x is the amount the spring is bent from equilibrium

Based on Hooke’s Law A force is required to bend a spring The more force applied the bigger the bend The more bend, the more strain energy stored

Force (N)

x (m)

Relationship between F and xslope of the line = k

Area under curve is

the amount of stored

strain energy

Stored strain energy = ½ k x2

Hooke’s Law from a Graphical Point of

View

x(m) Force(N)

0 0

0.1 12

0.2 24

0.3 36

0.4 48

0.5 60

0.6 72

graph x vs.F a of Slope

k

x

Fk

kxF

s

s

Force vs. Displacement y = 120x + 1E-14

R2 = 1

0

10

20

30

40

50

60

70

80

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Displacement(Meters)

Fo

rce(N

ew

ton

s)

Suppose we had the following data:

k =120 N/m

ELASTIC POTENTIAL ENERGY

Recall that the force of an elastic spring is F = ks. It

is important to realize that the potential energy of a

spring, while it looks similar, is a different formula.

Notice that the potential

function Ve always yields

positive energy.

2

2

1ksVe

Ve (where ‘e’ denotes an elastic

spring) has the distance “s”

raised to a power (the result of

an integration) or

Elastic potential energy

2

0

2

00

21|

2|

)(

)()(

kxUWx

kW

dxxkdxkxW

dxkxdxxFW

spring

x

x

x

x

x

x

Elastic “potential” energy is a fitting term as

springs STORE energy when there are

elongated or compressed.

Conservation of Energy in

Springs

ExampleA slingshot consists of a light leather cup, containing a stone, that is pulled back against 2 rubber bands. It takes a force of 30 N to stretch the bands 1.0 cm (a) What is the potential energy stored in the bands when a 50.0 g stone is placed in the cup and pulled back 0.20 m from the equilibrium position? (b) With what speed does it leave the slingshot?

v

vmvU

KUEEc

kkxUb

kkkxFa

s

sAB

s

s

22

2

)050.0(2

12

1

)

)20)(.(5.02

1)

)01.0(30) 3000 N/m

300 J

109.54 m/s