Chapter 12 – Simple Chapter 12 – Simple MachinesMachines

A PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of Paul E. Tippens, Professor of PhysicsPhysics

Southern Polytechnic State Southern Polytechnic State UniversityUniversity© 2007

SIMPLE MACHINESSIMPLE MACHINES are used to perform a variety are used to perform a variety of tasks with considerable efficiency. In this of tasks with considerable efficiency. In this example, a system of gears, pulleys, and levers example, a system of gears, pulleys, and levers function to produce accurate time measurements.function to produce accurate time measurements.

Photo Vol. 1 Photo Vol. 1 PhotoDisk/GettyPhotoDisk/Getty

Objectives: After completing Objectives: After completing this module, you should be this module, you should be able to:able to:• Describe a Describe a simple machinesimple machine in general terms and in general terms and

apply the concepts of apply the concepts of efficiencyefficiency, , energy energy conservationconservation, , workwork, and , and powerpower..

• Distinguish by definition and example between the Distinguish by definition and example between the concepts of the concepts of the idealideal and and actual mechanical actual mechanical advantages.advantages.

• Describe and apply formulas for the mechanical Describe and apply formulas for the mechanical advantage and efficiency of the following devices: advantage and efficiency of the following devices: (a) (a) leverslevers, (b) , (b) inclined planesinclined planes, (c) , (c) wedgeswedges, (d) , (d) gearsgears, , (e) (e) pulley systemspulley systems, (f) , (f) wheel and axelwheel and axel, (g) , (g) screw screw jacksjacks, and (h) the , and (h) the belt drivebelt drive..

A Simple MachineA Simple Machine

In a In a simple machinesimple machine, , input workinput work is done is done by the application of by the application of a single force, and a single force, and the machine does the machine does output workoutput work by by means of a single means of a single force.force.

Conservation of energy Conservation of energy demands that the demands that the work input be equal to the sum of the work input be equal to the sum of the work output and the heat lost to friction.work output and the heat lost to friction.

A simple A simple machinemachine

ssinin

ssoutout

WWinin= F= Fininssinin

WWoutout= = FFoutoutssoutout

FFinin

FFoutout

WW

A Simple Machine (Cont.)A Simple Machine (Cont.)

A simple A simple machinemachine

ssinin

ssoutout

WW

FFoutout

FFininFFinin

WW

FFoutout

FFinin

WWinin= F= Fininssinin

WWoutout= F= Foutoutssoutout

Input work = output work + work against friction

Input work = output work + work against friction

Efficiency Efficiency ee is is defined as the ratio defined as the ratio of work output to of work output to work input.work input.

out out

in in

Work outpute

Work input

F se

F s

Example 1.Example 1. The The efficiency of a simple efficiency of a simple machine is machine is 80%80% and a and a 400-N 400-N weight is lifted a weight is lifted a vertical height of vertical height of 2 m2 m. . If an input force of If an input force of 20 N20 N is required, what is required, what distance must be distance must be covered by the input covered by the input force?force?The efficiency is 80% or The efficiency is 80% or ee = 0.80, = 0.80, thereforetherefore

or out out out outin

in in in

F s F se s

F s eF

(400 N)(2 m)

(0.80)(20 N)ins sin = 5.0 m

A A simple simple machinmachin

ee

ssinin

ssoutout

WW

FFinin = ? = ?

WW out out

in in

F se

F s

EfficiencEfficiencyy

The advantage is a reduced input force, but it is at the expense of

distance. The input force must move a greater distance.

Power and EfficiencyPower and EfficiencySince power is work per Since power is work per unit time, we may writeunit time, we may write

0out

in i

W Pte

W Pt

A A simple simple machinmachin

ee

ssinin

ssoutout

WW

FFinin = ? = ?

WW out

in

Pe

P

EfficiencEfficiencyy

or Work Work

P Ptt

0

i

Pe

P

0

i

PPower oute

Power in P

Efficiency is the Efficiency is the ratio of the power ratio of the power

output to the output to the power input.power input.

Example 2.Example 2. A12-hpA12-hp winch motor lifts a winch motor lifts a 900-lb900-lb loadload

Po = 6270 ftlb/s

A A simple simple machinmachin

ee

ssinin

ssoutout

WW

FFinin = ? = ?

WW out

in

Pe

P

EfficiencEfficiencyy

First we must First we must find the find the power power output, output, PPoo::

0

i

Pe

P

to a height of to a height of 8 ft8 ft. What . What is the output power in is the output power in ftftlb/slb/s if the winch is if the winch is 95%95% efficient? efficient?

PPoo = (0.95)(12 hp) = 11.4 hp= (0.95)(12 hp) = 11.4 hp

(1 hp = 550 (1 hp = 550 ft/s):ft/s):

550 ft lb/s(11.4 hp) 6600 ft lb/s

1 hpoP

0 iP eP

Ex. 2 (cont.)Ex. 2 (cont.) A12-hpA12-hp winch motor lifts a winch motor lifts a 900 900 lblb load load

A A simple simple machinmachin

ee

ssinin

ssoutout

WW

FFinin = ? = ?

WW out

in

Pe

P

EfficiencEfficiencyy

We just found We just found that Pthat Poo = 6270 W = 6270 W

to a height of 8 ft. How to a height of 8 ft. How much time is required if much time is required if the winch is the winch is 95%95% efficient?efficient?

o oo

F sWork outP

t t

(900 lb)(8 ft)

6270 o o

o

F st

P Now we solve for Now we solve for t t

::

Time required: t = 1.15 sTime required: t = 1.15 s

Actual Actual Mechanical Mechanical AdvantageAdvantage

A simple A simple machinemachine

ssinin

ssoutout

WW

FFinin = ? = ?

WW

Actual Actual MechanicaMechanica

l l AdvantageAdvantage

FFoutout

MMAA

The The actual actual mechanical mechanical advantageadvantage,, M MAA, is the , is the ratio of ratio of FFoo to to FFii..

o

Ai

Foutput forceM

input force F

80 N80 N

40 N40 NFor example, if an For example, if an input force of input force of 40 N40 N lifts an lifts an 80 N80 N weight, weight, the actual the actual mechanical mechanical advantage is:advantage is:

80 N

40 N2.0

A

A

M

M

An Ideal MachineAn Ideal MachineConservation of energy demands that:Conservation of energy demands that:

Input work = output work + work against friction

Input work = output work + work against friction

( )i i o o fFs F s Work AnAn idealideal or perfect machine is 100% or perfect machine is 100%

efficient and (Work)efficient and (Work)ff = 0, so that = 0, so that

or o ii i o o

i o

F sF s F s

s s

The ratio The ratio ssii/s/soo is the is the ideal ideal mechanical mechanical advantage.advantage.

Ideal Ideal Mechanical Mechanical AdvantageAdvantage

A simple A simple machinemachine

ssinin

ssoutout

WW

FFinin = ? = ?

WW

Ideal Ideal MechanicaMechanica

l l AdvantageAdvantage

FFoutout

MMII

The The ideal mechanical ideal mechanical advantageadvantage,, MMII, is the , is the ratio of ratio of ssin in to to ssoutout..

i

Ao

sin distanceM

out distance s

For example, if anFor example, if an input input force moves a distance of force moves a distance of 6 6 mm while the while the outputoutput force force moves moves 2 m2 m, the ideal , the ideal mechanical advantage is:mechanical advantage is:

6 m

2 m3.0

I

I

M

M

6 m6 m

2 m2 m

Efficiency for an Ideal EngineEfficiency for an Ideal Engine

For 100% efficiency For 100% efficiency MMAA = M = MII. In other . In other words, in the absence of friction, the words, in the absence of friction, the machine IS an ideal machine and machine IS an ideal machine and ee = 1. = 1.

A A simple simple machimachi

neneSSin in = 8 = 8

mmSSoutout= 2 = 2

mmWW

FFinin = 80 N = 80 N

WW ee = = 100%100%

FFoutout==

400 N400 N

IDEAL IDEAL EXAMPLE:EXAMPLE:

8 m4

2 mi

Io

sM

s

80 N4

20 No

Ai

FM

F

1.0A

i

Me

M

Efficiency for an Actual EngineEfficiency for an Actual Engine

The actual efficiency is always less than The actual efficiency is always less than the ideal efficiency because friction the ideal efficiency because friction always exits. The efficiency is still equal always exits. The efficiency is still equal to the ratio to the ratio MMAA/M/MII..

A

i

Me

MThe efficiency of

any engine is given by:

In our previous example, the ideal In our previous example, the ideal mechanical advantage was equal to mechanical advantage was equal to 44. If . If the engine was only the engine was only 50% efficient50% efficient, the , the actual mechanical advantage would be actual mechanical advantage would be 0.5(4) or 0.5(4) or 22. Then . Then 160 N160 N (instead of 80 N) (instead of 80 N) would be needed to lift the 400-N weight.would be needed to lift the 400-N weight.

In our previous example, the ideal In our previous example, the ideal mechanical advantage was equal to mechanical advantage was equal to 44. If . If the engine was only the engine was only 50% efficient50% efficient, the , the actual mechanical advantage would be actual mechanical advantage would be 0.5(4) or 0.5(4) or 22. Then . Then 160 N160 N (instead of 80 N) (instead of 80 N) would be needed to lift the 400-N weight.would be needed to lift the 400-N weight.

The LeverThe Lever

The input torque The input torque FFiirri i is is equal to the output equal to the output torque torque FFoorroo..

i i o oFr F r

The actual mechanical advantage is, therefore:

o iA

i o

F rM

F r

A A leverlever shown here shown here consists of input and consists of input and output forces at output forces at different distances different distances from a fulcrum.from a fulcrum.

FFinin

FFoutout

rroutout rrinin

FulcrumFulcrum

The LeverThe Lever

Friction is Friction is negligible so that negligible so that

WWoutout = W = Winin::

or oo

ii

io

i

F

ss

FF F

ss

The ideal MI is: and o iI I A

i o

F rM M M

F r

Note from figure thatNote from figure that angles are the same angles are the same and arc length and arc length ss is proportional to is proportional to rr. . Thus, Thus, the ideal mechanical advantage is the the ideal mechanical advantage is the same as actual.same as actual.

FFinin

FFoutout rroutout

ssoutout

ssinin

rrinin

Example 3.Example 3. A A 1-m1-m metal lever is used to lift metal lever is used to lift a a 800-N 800-N rock. What force is required at the rock. What force is required at the left end if the fulcrum is placed left end if the fulcrum is placed 20 cm20 cm from from the rock?the rock?

rriirr22

800 N800 N

F = ?F = ?

1. Draw and label 1. Draw and label sketch:sketch:2. List given info:2. List given info:

FFoo = = 700 N; 700 N; rr22 = = 20 20 cmcm

rr11 = = 100 cm - 20 cm = 80 100 cm - 20 cm = 80 cmcm

3. To find 3. To find FFii we recall the definition of M we recall the definition of MII : :80 cm

and 4 ;20 cm

iI I

o

rM M

r For lever:For lever: MMAA = M = MII

4oA

i

FM

F Thus,Thus, 800 N

200 N4iF andand

Other Examples of LeversOther Examples of Levers

Wheel and Wheel and Axel:Axel:

RRrr

FFii

FFoo

Wheel and AxelWheel and Axel

Application of Lever Application of Lever Principle:Principle:With no friction With no friction MMII = M = MAA

andandFor Wheel and Axel:

o iA

i o

F rM

F r

For example, if For example, if R = 30 cmR = 30 cm and and r = 10 cmr = 10 cm, , an input force of only an input force of only 100 N100 N will lift a will lift a 300-300-NN weight! weight!

If the smaller radius is 1/3 of the larger If the smaller radius is 1/3 of the larger radius, your output force is 3 times the radius, your output force is 3 times the input force.input force.

If the smaller radius is 1/3 of the larger If the smaller radius is 1/3 of the larger radius, your output force is 3 times the radius, your output force is 3 times the input force.input force.

Single Fixed PulleysSingle Fixed PulleysSingle fixed pulleys serve only to change Single fixed pulleys serve only to change

the direction of the input force. See the direction of the input force. See examples:examples:

W

FFininFFoutout

FFinin

FFoutout

FFin in = F= Foutout

Single Moveable PulleySingle Moveable Pulley

80 N80 N

FFinin

A free-body diagram shows an A free-body diagram shows an actualactual mechanical advantagemechanical advantage of of MMAA = = 2 2 for a single for a single

moveable pulley.moveable pulley.2 or 2o

in out Ai

FF F M

F

FFinin + F + Fin in = = FFoutout

40 N + 40 N40 N + 40 N = = 80 80 NN80 N80 N

FFinin

FFoutout

FFinin

1 m1 m2 m2 m

Note that the rope moves a distance of 2 m while the weight is

lifted only 1 m.

2inI

out

sM

s

Block and Tackle Block and Tackle ArrangementArrangement

W FFoo

FFii

We draw a free-body We draw a free-body diagram:diagram:

FFoo

FFii

FFii FFii FFii

The lifter must pull 4 m of rope in order to lift the weight 1 m

4

4

in out

oA

i

F F

FM

F

A A belt drivebelt drive is a device used to transmit is a device used to transmit torque from one place to another. The torque from one place to another. The actual mechanical advantage is the ratio of actual mechanical advantage is the ratio of the torques.the torques.

The Belt DriveThe Belt Drive

Belt Belt DriveDrive

FFoo

FFii

o

Ai

output torqueM

input torque

o

Ai

output torqueM

input torque

Since torque is defined as Since torque is defined as FrFr, the ideal advantage is:, the ideal advantage is:

o oI A

i i

F rM M

Fr

o oI

i i

r DM

r D Belt Drive:

rrii

rroo

Angular Speed RatioAngular Speed RatioThe mechanical The mechanical advantage of a belt drive advantage of a belt drive can also be expressed in can also be expressed in terms of the diameters terms of the diameters DD or in terms of the or in terms of the angular speeds angular speeds ..

o iI

i o

DM

D

Belt Drive:

Note that the smaller Note that the smaller pulley diameter always pulley diameter always has the greater has the greater rotational speed.rotational speed.

Belt Belt DriveDrive

DDoo

DDii

Speed Speed ratio:ratio:

i

o

Example 4.Example 4. A A 200 N200 Nmm torque is applied to an torque is applied to an input pulleyinput pulley 12 cm12 cm in in diameter. (a) What should diameter. (a) What should be the diameter of the be the diameter of the output pulley to give an output pulley to give an ideal mechanical ideal mechanical advantage of advantage of 44? (b) What ? (b) What is the belt tension?is the belt tension?

MMII = 4 = 4

FFoo

FF

rrii

rroo

To find DTo find Doo we use the fact we use the fact thatthat 4; 4o

I o ii

DM D D

D

DDoo = 4(12 cm) = = 4(12 cm) = 48 48 cmcmNow, Now, i i = = FFiirri i and and rri i = =

DDii/2. Belt tension is /2. Belt tension is FFi i and and rrii is equal to ½D is equal to ½Dii = = 0.06 m.0.06 m.

200 N mi i iFr

200 N m

0.06 3 N

m3 30iF

oI

i

DM

D

GearsGears

o oI

i i

D NM

D N Gears:

In this case, Do is the diameter of the driving gear and Di is diameter of the driven gear. N is the number of teeth.

NiNo

If 200 teeth are in the input (driving) gear, and 100 teeth in the output (driven) gear, the mech-anical advantage is ½.

Mechanical Mechanical advantage of gears advantage of gears is similar to that for is similar to that for belt drive:belt drive:

Example 5.Example 5. The driving gear on a bicycle The driving gear on a bicycle has has 4040 teeth and the wheel gear has only teeth and the wheel gear has only 2020 teeth. What is the mechanical teeth. What is the mechanical advantage? If the driving gear makes advantage? If the driving gear makes 60 60 rev/minrev/min, what is the rotational speed of the , what is the rotational speed of the rear wheel? rear wheel?

22; 0.5

44o

I Ii

NM M

N

Remember that the Remember that the angular speed ratio is angular speed ratio is opposite to the gear opposite to the gear

ratio.ratio. 1;

2o i i

Ii o o

NM

N

o o = 2= 22(60 2(60 rpm)rpm)

Output angular speed:

= 120 rpm

Output angular speed:

= 120 rpm

Ni = 40No = 20

The Inclined PlaneThe Inclined Plane

FFii

FFo o = = WW

ssoo

ssii

The Inclined The Inclined PlanePlane

Ideal Mechanical Ideal Mechanical AdvantageAdvantage

iI

o

sslopeM

height s

Actual Advantage: Ai

WM

F

Because of friction, the actual mechanical advantage MA of an inclined plane is usually much less than the ideal mechanical advantage MI.

Because of friction, the actual mechanical advantage MA of an inclined plane is usually much less than the ideal mechanical advantage MI.

Example 6.Example 6. An inclined plane has a slope An inclined plane has a slope of of 8 m8 m and a height of and a height of 2 m2 m. What is the . What is the ideal mechan-ical advantage and what is ideal mechan-ical advantage and what is the necessary input force needed to push the necessary input force needed to push a a 400-N 400-N weight up the incline? The weight up the incline? The efficiency is efficiency is 60 percent60 percent..

FFii

FFo o = 400 = 400 NN

2 2 mm

SSi i = = 8 8 mm

8 m;

2 m 4 i

I Io

sM

sM

; (0.60)(4)AA I

I

Me M eM

M

2.4 oA

i

FM

F 400 N

2.4 2.4o

i

FF Fi = 167

N

Fi = 167 N

The Screw JackThe Screw Jack

p

Fo

FiR

Screw Jack

2I

RM

p

An application of the An application of the inclined plane:inclined plane:

Input distance: Input distance: ssii = = 22R R Output distance: Output distance: ssoo = = pp

2iI

o

Screw Jack

s RM

s p

Due to friction, the screw jack is an Due to friction, the screw jack is an inefficientinefficient machine with an actual machine with an actual mechanical advantage significantly mechanical advantage significantly lessless than the ideal advantage.than the ideal advantage.

Summary for Simple Summary for Simple MachinesMachines

0

i

PPower oute

Power in P

Efficiency is the Efficiency is the ratio of the power ratio of the power

output to the output to the power input.power input.

EfficiencyEfficiency e e is is defined as the ratio defined as the ratio of work output to of work output to

work input.work input.

out out

in in

Work outpute

Work input

F se

F s

SummarySummary

The The ideal mechanical ideal mechanical advantageadvantage,, MMII, is the , is the

ratio of ratio of ssin in to to ssoutout..

i

Ao

sin distanceM

out distance s

The The actual actual mechanical mechanical

advantageadvantage,, M MAA, is the , is the ratio of ratio of FFoo to to FFii..

o

Ai

Foutput forceM

input force F

A A simple simple machinmachin

ee

ssinin

ssoutout

WW

FFinin = ? = ?

WW out

in

Pe

P

EfficiencEfficiencyy

Summary (Cont.)Summary (Cont.)

The actual mechanical advantage for a lever:

o iA

i o

F rM

F r

Application of lever Application of lever principle:principle:With no friction With no friction MMII = M = MAA

For Wheel and axel:

o iA

i o

F rM

F r

Summary (Cont.)Summary (Cont.)

o

Ai

output torqueM

input torque

o

Ai

output torqueM

input torque

o oI

i i

r DM

r D Belt Drive:

o iI

i o

DM

D

Belt Drive:

MMII = 4 = 4

FFoo

FF

rrii

rroo

Belt Belt DriveDrive

SummarySummary

o oI

i i

D NM

D N Gears:

FFii

FFo o = W= W

ssoo

ssii

The Inclined The Inclined PlanePlane

Ideal Mechanical Ideal Mechanical AdvantageAdvantage

iI

o

sslopeM

height s

Actual Advantage: Ai

WM

F

NiNo

Summary (Cont.)Summary (Cont.)

p

Fo

FiR

Screw Jack

2I

RM

p

An application of the An application of the inclined plane:inclined plane:

Input distance: Input distance: ssii = = 22R R Output distance: Output distance: ssoo = = pp

2iI

o

Screw Jack

s RM

s p

CONCLUSION: Chapter 12 CONCLUSION: Chapter 12 Simple MachinesSimple Machines