Work, Power, and Machines

Work• Push a box across the floor. • When a force is exerted over a distance, work is done.• Work is the product of force and distance when the two

are in the same direction.W Fd

d

F

Sample Work Problem• What is the work done on the box if the force exerted is

44N and the distance covered is 3.5m?

d = 3.5m

F = 44N

W Fd 44 3.5W N m154W J

Work is measured in Joules (J), the unit for energy.

Practice Problem 1 (Work)• A smaller box is pushed across the same floor.• The force required here is 20N and the distance covered

is again 3.5m.• What is the work done? W Fd

20 3.5W N m70W J

Practice Problem 2 (Work)• Work can be done against gravity when objects are lifted.

• The force required to lift something is equal to its weight.

• How much work is done to lift the weight to the 2m shelf?

3m

2m

1m

10KgF d

F mg 210 9.8m

sF kg

98F N

W Fd 98 2W N m196W J

0

Now Find Work Done.

Find Weight (Force). Lift

Sample Work Problem 2• The box (m = 15kg) is now lifted a height of 1.5m.• How much work is done?

The force required to lift an object is equal to its weight.

1.5m

F mg 15 9.8F kg147F NcosW Fd

147 1.5 cos 0W N m 220.5W J

F

Lift

d

Work and Direction• There are occasions in which forces are not in the same

direction as the distance covered (displacement). • Consider a sled being dragged across the snow by a

diagonal rope.

• The angle between the vector

quantities of force and displacement

is denoted as

What is the direction of motion of the sled?

What is the direction of the force acting on the sled?

d

F

Work and Direction (cont.)• We are only interested in the component of force in the

direction of the displacement (Fd).

• A trig function must be introduced in order to calculate work when directions are different.

• Now the sled problem is solvable!

d

F

Make sure your calculator is in degree mode!

F

Fd

cos dF

F cosdF F cosW Fd

dW F d

Fd

Sample Work Problem 3• A sled is dragged across 45m snow with a force of 33N

being exerted on the string, which is at an angle of 23°.• How much work is being done on the sled?

d

F

cosW Fd 33 45 cos 23W N m 1367W J

Practice Problem 2 (Work)• A Lawnmower is being pushed across the grass. The

angle of the handle is 50° with the horizontal. The job requires the a force of 80N across the 33m yard.

• How much work is done?

F

d50

cosW Fd (80 )(33 )cos50W N m 1697W J

50

When No Work Is Done• Carry a box horizontally across the room.• What is the direction of motion?• What is the direction of the force?• What is the angle ?• What is cos90 ?

cosW Fd

0

So when force and displacement vectors are perpendicular, no work is done.

d

F

90

Work and Path• Consider the movements of the three identical balls below.

Each starts from the ground and ends up at the entry doors.

• Which ball requires the most work if there is no friction present?

• The answer is that they all require the same amount of work because portions of their paths may have required less work or even no work, but the totals were equal.

Roll

Work And Friction• Friction requires work to be done.

• Consider a box being dragged across a sidewalk.

• What is the direction of the displacement?

• What is the direction of the frictional force?

• What is the angle between the vectors below?

• What is cos 180°?

• This means that work can be negative.

• This may include situations other than friction.

d F 180

1

60 44 cos 180W N m 2640W J

Practice Problem 3 (Work)• Remember the box that was dropped from a moving truck?

• Whatever happened to that box anyway?

• The box slides 44m before coming to a stop, experiencing a force of friction of 60N.

• What is the work done by friction?

Drop

d F

cosW Fd

Which Does More Work• After waking up on a cold winter morning, you find your

driveway blanketed with snow.• You have two choices for clearing the snow.

– A snow shovel (Exerts 200N over a distance of 8m in 25s)– Or a snow blower (Exerts 200N over a distance of 8m in 10s)

• Which way involves more work in removing the snow?• The answer is that they do the same amount of work

because the same force is exerted over the same distance. The power is different however.

Power• Power is the rate at which work is being done.• The unit for power is the Watt (W).• The equation for power is shown below.

WP

t

Power (W)

Time (s)

Work (J)

Calculating Power• Now let’s calculate the power of the snow

shovel against that of the snow blower.

WP

t

WP

t

1600

25

JP

s 1600

10

JP

s

64P W 160P W

Snow Shovel Snow Blower

Sample Power Problem• An electric motor can do 4000J of work in a time of 8s.• How much is the power provided by the engine?

WP

t

4000

8

JP

s

500P W

Practice Problem 4 (Power)• Another electric motor is connected with a belt and pulley to a

grinding machine.• The motor exerts a force of 340N and turns a distance of

10m. The entire process takes 17s.• Assuming no energy losses, what was the work done by the

motor? What is the power that the motor uses?

WP

t

3400

17

JP

s 200P W

340 10 cos 0W N m cosW Fd

3400W J

Grind

Simple Machines• Machines are designed to make our lives easier. • In terms of forces, they can change direction, magnitude, or both.• However, we must remember that machines still require the

same amount of work (and in some cases more) to be done.• Any combination of these would constitute a compound machine.

F

LE

Lever Pulley Incline Plane

Wedge Wheel & Axle Screw

Mechanical Advantage• Mechanical advantage is the ratio of output force to input force.• This shows how many times a force is multiplied by a machine.

– If MA = 1, then no forces are multiplied.– If MA > 1, then forces are multiplied.– If MA < 1, then forces are divided.

• The trade off is distance. However many

times the forces are multiplied, so the

Input distance exceeds the output distance.• Mechanical advantage is unitless.

O

I

FMA

F

500-g

FO

FIdI

dO

Sample Problem (MA)• You move a large rock (m = 400kg) using a bar by

exerting a force of 490N on the bar.• What is the mechanical advantage of the bar lever?

(The output force is the rock weight in this problem.)Lift

O

I

FMA

F

3920

490

NMA

N 8MA

OF mg 2400 9.8mO s

F kg3920OF N

400kg

Ideal Mechanical Advantage• The ideal mechanical advantage (IMA) of a machine is

the ratio of input distance (dI) to output distance (dO).

• IMA shows what the mechanical advantage would be in a perfect situation.

• Friction and other energy losses often cause the MA to be less than ideal.

• Like MA, IMA is also unitless.

I

O

dIMA

d

Sample Problem (IMA)• A bottle jack is used to lift heavy objects.• For each 0.2m push of the operator, the piston rises a

distance of 0.01m.• What is the IMA of the jack? Jack

I

O

dIMA

d

0.220

0.01

mIMA

m

Levers• A lever consists of a rigid shaft that pivots about

a fixed point.• Each lever is made up of three major parts:

– Fulcrum – Pivot Point

– Effort – Input Force (FI)

– Load (Resistance) – Output Force (FO)

• There are three major types of levers.

E

F

L

Lever Types• There are three major types of levers.

• This depends on what component is located in the center.– Type I – Fulcrum in Center– Type II – Load in Center– Type III – Effort in Center

F

LE

Type I

F

L

E

Type II

EF

L

Type III

Examples of Type I Levers• These are also called “1st Class Levers”

(Fulcrum in Center)

Seesaw

Pry Bar

Pliers/Scissors

Examples of Type II Levers• These are also called “2nd Class Levers”

(Load in Center)

Bottle Jack

Wheelbarrow

Nutcracker

Examples of Type III Levers• These are also called “3rd Class Levers”

(Effort in Center)

Tennis Racket

Baseball Bat

Golf Club

Solving Lever Problems• Input and output distance (dI and dO) are found by finding

how far the effort (FI) and load (FO) are from the fulcrum.

• The following relationship exists in a perfect lever:

• This applies to all three types.

F

L E

dO dI

I I O OF d F d

(FO) (FI)

Sample Lever Problem• You fill a wheel barrow with 30kg of pumpkins. The load

is centered 0.35m behind the wheel. The handles are located 1.15m behind the wheel. What force is required to lift the load?

Pullies• A pulley is a grooved wheel

that can be used to manipulate the force of a rope or cable.

• Pulleys come in 2 types:– Fixed (redirects force)– Moveable (multiplies force)

Fixed Pulley

Moveable Pulley

Pulleys (Example 1) • The approximate MA for a pulley system is

the number of supporting strands.

Pulleys (Example 2)• The approximate MA for a pulley system is

the number of supporting strands.

Pulleys (Example 3) • The approximate MA for a pulley system is

the number of supporting strands.

Pulleys (Example 4) • The approximate MA for a pulley system is

the number of supporting strands.

Inclined Plane

Efficiency• Efficiency is the percentage of work input (WI)

successfully converted into work output (WO) by a machine.

• The equation can also be found in two other useful forms.

100O

I

Weff

W

100O O

I I

F deff

F d

100MA

effIMA

Sample Efficiency Problem 1• An axe (a wedge) is used to split a piece of wood, a job

requiring 405J of work. The person swinging the axe does 516J. What is the efficiency of the axe?

100O

I

Weff

W

405100

516

Jeff

J

78.5%eff

Sample Efficiency Problem 2• A pulley system is used to lift a cannon (400kg) onto a

boat 11m high. The pulley system requires a person to pull with a force of 560N over a distance of 88m. What is the efficiency of the machine?

Input Work

Output Work

Efficiency

O O OW F d 3920 11OW N m43,120OW J

I I IW F d 560 88IW N m49,280IW J

100O

I

Weff

W

43,120100

49,280

Jeff

J

87.5%eff