WORK, ENERGY, AND SIMPLE MACHINES

Work - the product of the force exerted on an object

and the distance the object moves in the direction

of the force.

W = F d cos θ

W: work

F: magnitude of force

d: magnitude of displacement

Units are Joules (J = Nm)

If a 1 N force moves an object 1 m, 1 Joule of work is

done.

Work is only done on an object if the object moves.

Work is only done on an object when the force and

displacement are in the same direction.

Example. Holding an object at the same height for

hours - you may get tired but you are doing no work on

the object. Even if you are moving and the object

remains at the same height, no work is being done.

What is the effect of doing work?

In simple terms, the work done on an object is equal to

the energy of motion or kinetic energy.

2

21cos vmEFdW k ∆=∆⇒= θ

What about the time it takes to move an object?

Power is the rate of doing work.

Power is the rate at which energy is transferred.

t

WP = Units: Watts (W)

If you have a F vs d graph,

the area under the curve represents work.

Remember, work is done if a force is exerted in the

direction of motion.

If you are pushing or pulling something at an angle, only

the component that acts in the direction of motion is

doing work.

W = F d cos Θ

Example. Lawn mower. A person is doing work on the

lawn mower. The lawn exerts a force (friction) on the

lawn mower. W = Ffriction d cos Θ

F (N)

d (m)

Ex. 1 An applied force of 20. N accelerates a block

across a level, frictionless surface from rest to a

velocity of 8.0 m/s in a time of 2.5 s. Calculate the

work done by this force.

Ex. 2 Calculate the work done in lifting a 12 kg crate

from the floor to a platform 3.0 m above floor level at

a constant velocity.

Ex. 3 A truck pushes a car by exerting a horizontal

force of 500. N on it. A frictional force of 300. N

opposes the car’s motion as it moves 4.0 m. Calculate

the work done on the car by the truck.

Ex. 4 Calculate the work done by a horse that exerts

an applied force of 100. N on a sleigh, if the harness

makes an angle of 30.o with the ground, and the sleigh

moves 30. m across a flat, level ice surface.

Ex. 5 A 50. kg crate is pulled 40. m along a horizontal

floor by a constant force exerted by a person,

Fp = 100. N, which acts at 37o angle. The floor is rough

and exerts a friction force Ff = 50. N. Determine the

work done by each force acting on the crate, and the

net work done on the crate.

Ex. 6 Determine the work a person must do to carry a

15.0 kg backpack up a hill of height 10.0 m. Determine

also the work done by gravity and the net work done on

the backpack. Assume the motion is smooth and at a

constant velocity.

Machines

- whether powered by people or engines, machines

make our tasks easier.

- a machine eases the load either by changing the

magnitude or direction of the force. It does not

change the amount of work done.

Example. A bottle opener. You lift the handle of the

opener and it lifts the cap off.

The work you do to lift the handle is the input work,

Wi.

The work the machine does is the output work, Wo.

Work is the transfer of energy by mechanical means.

The output work can never be greater than the input

work. The machine aids in the transfer of energy from

you to the cap. The cap cannot receive more energy

than you put into the machine.

The force you exert on a machine is the effort force,

Fe.

The force exerted by the machine is the resistance

force, Fr.

The mechanical advantage of the machine, AMA (or

MA) is:

e

r

F

FMA =

the ratio of the resistance force to the effort force.

Many machines will exert a MA greater than 1 because

it decreases the force you apply.

In terms of work,

rro dFW =

eei dFW =

oi WW =

rree dFdF =

r

e

e

r

d

d

F

F=

MA in terms of force

Ideal mechanical advantage is found by using the

measured distances moved.

Actual mechanical advantage is found using the

measured forces exerted.

In a real machine, some input work is lost and

therefore, not equal to the output work.

- losses from energy transformed to heat..

%100

%100

xIMA

MAefficiency

xW

Wefficiency

i

o

=

=

All machines are made from six simple machines.

These are *pulleys, *levers, wheel and axle, screws,

wedges, and *incline planes.

Pulleys

To calculate the ideal mechanical advantage of a pulley,

you count the number of supporting ropes except the

effort rope if it is being pulled down.

MA =

Incline Plane

To find the IMA of an incline plane,

R

e

d

dIMA =

de

dR

Levers

To determine the IMA of a lever,

R

e

d

dIMA =

Levers are classified as first, second, or third class.

First class: lever with pivot between the load and

effort.

Second class: lever with load between pivot and effort.

Third class: lever with effort between pivot and load.

scissors nut crackers fishing rod

Wedge

It is a double incline plane used to split something.

Wheel and Axle

The MA can be found by knowing the radius of the

wheel and the radius of the axle.

axle

wheel

e

r

r

r

F

FMA ==

Screw

To find the IMA, the radius of the bit and the

distance between each thread is needed.

pitch

rIMA

2=

Ex.1. A sledge hammer is used to drive a wedge into a

log to split it. When the wedge is driven 20. cm into

the log, the log is separated a distance of 5.0 cm. A

force of 1.9x104 N is needed to split the log and the

sledge exerts a force of 9.8x103 N.

a) Find IMA

b) Find MA

c) Find efficiency

Ex. 2. A worker uses a pulley system to raise a 225 N

carton 16.5 m. A force of 129 N is exerted on the

rope and the rope is pulled 33.0 m.

a) Find MA

b) Find efficiency

Ex.3. A boy exerts a force of 225 N on a lever to raise

a 1.25x103 N rock a distance of 0.13 m. If the

efficiency of the lever is 88.7%, how far did the

boy move his end of the lever?

Energy

Doing work means you have expended energy doing

something.

In physics, that expended energy means a force was

applied.

There is not much difference between work and

energy.

• In order to do work, an object must have energy • In order to have energy, an object must have work done on it.

ENERGY - the measure of a system's ability to do

work. Energy is classified into two terms, potential

energy and kinetic energy.

Potential energy - the energy stored in a body or

system as a consequence of its position, shape, or

form.

Kinetic energy - energy of motion.

Work is the transfer of energy by mechanical means.

When work is done, energy can be transferred from

potential to kinetic energy (kicking a football).

Kinetic energy:

2

2

1mvKE =

You can give an object more kinetic energy by

doing more work on it. In fact, W = ∆KE

Not all objects are at rest. They may already have

kinetic energy when additional work is done on

them.

KEKEKEW ifnet ∆=−=

Work-energy theorem: the net work done on an object

is equal to its change in kinetic and potential energies.

Potential Energy:

mghPE =

Energy is transferred from kinetic to potential and

back to kinetic. We describe the total energy, E, as:

KEPEET +=

The height of where an object is measured from is the

reference level.

Conservation of Energy

Law of Conservation of Energy: within a closed,

isolated system, energy can change form, but the

total amount of energy is constant.

- energy can neither be created or destroyed.

21 EE =

2211 PEKEPEKE +=+

Ex. 1. A 145 g baseball is thrown with a speed of 25

m/s.

a) What is its KE?

b) How much work was done to reach this speed

starting from rest?

Ex.2. How much work is required to accelerate a

1000. kg car from 20. m/a to 30. m/s?

Ex. 3. A 1000. kg car moves from point A to point B

and then point C. The vertical distance between A

and B is 10. m and between A and C is 15. m.

B

A

C

a) What is the PE at B and C relative to A?

b) What is the ∆PE when it goes from B to C? c) Repeat a) and b) but take the reference level at C.

Ex. 4. A roller coaster is shown below. Assuming

there is no friction, calculate the speeds at points

B,C, and D if it has a speed of 1.8 m/s at A.

A

B

C

D

30. m 25 m 12 m

Ex. 5 (friction) A girl has a mass of 28 kg. She

climbs a 4.8 m ladder of a slide, and reaches a

velocity of 3.2 m/s at the bottom of the slide. How

much work was done by friction on the girl?

Ex. 6 A 2.75 kg box is at the top of a frictionless

incline. What is the potential energy of the box with

respect to the bottom of the incline?

Ex. 7 A heavy box slides down a frictionless incline.

If the box starts from rest at the top of the incline,

what is its speed at the bottom?

10.0 m

7.00 m

12.0 m

30.0o

Elastic Potential Energy

Many objects can stretch, compress, or change shape

in many ways, but if it returns to its original

condition, it is said to be elastic.

When an object continues to move when the force is

removed, there must have been energy stored in the

object due to its condition. This is elastic potential energy.

Hooke’s Law

This explains the relationship between the extension

of a spring and the force exerted on it.

Springs can be stretched or compressed and the

force that will restore it to its original state is the

restoring force. It acts in the opposite direction to the direction that the spring compresses or

stretches.

The restoring force and displacement are

proportional, the graph represents a linear curve.

y = mx + b

The slope of the line (m) represents the spring

constant, k (N/m).

The spring constant is a measure of the stiffness of

the spring.

kxF

kxF

=−=

F: the restoring force (N) � -kx

F: the applied force (N) � kx

k: spring constant (N/m)

x: extension / displacement (m)

Area under the curve is

2

21

21

kxE

kxF

EFx

pe

pe

=

=

=

F (N)

X (m)

Area is the Ep

Ex. 1 A 0.25 kg mass hangs on the end of a spring. What is its

extension if it has a spring constant of 48 N/m?

Ex. 2 How much energy does a bow have when pulled back 8.0 cm if

it has a force constant of 160 N/m?