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Chapter 12 Chapter 12 –– Simple MachinesSimple MachinesA PowerPoint Presentation byA PowerPoint Presentation by

Paul E. Tippens, Professor of PhysicsPaul E. Tippens, Professor of Physics

Southern Polytechnic State UniversitySouthern Polytechnic State University

© 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 PhotoDiskPhotoDisk/Getty/Getty

Objectives: After completing this Objectives: After completing this module, you should be able to:module, you should be 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 jacksscrew jacks, 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 by is done by the application of a the application of a single force, and the single force, and the machine does machine does output output workwork by means of a by means of a single force.single force.

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

A simple A simple machinemachine

ss inin

ss outout

WWinin = = FFinin ssinin

WWoutout = = FFoutout ssoutout

FF inin

FF outout

WW

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

A simple A simple machinemachine

ssinin

ssoutout

WW

FFoutout

FFininFFinin

WW

FFoutout

FFinin

WWinin = = FFinin ssinin

WWoutout = = FFoutout ssoutout

Input work = output work + work against frictionInput work = output work + work against friction

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

out out

in in

Work outputeWork inputF seF s

Example 1.Example 1. The efficiency The efficiency of a simple machine is of a simple machine is 80%80% and a and a 400400--N N weight weight is lifted a vertical height of is lifted a 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 covered distance must be covered by the input force?by the input force?

The efficiency is 80% or The efficiency is 80% or ee = 0.80, therefore= 0.80, therefore

or out out out outin

in in in

F s F se sF s eF

(400 N)(2 m)(0.80)(20 N)ins sin = 5.0 m

A simple A simple machinemachine

ss inin

ss outout

WW

FF inin = ?= ?

WW out out

in in

F seF s

EfficiencyEfficiency

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 P teW Pt

A simple A simple machinemachine

ss inin

ss outout

WW

FF inin = ?= ?

WW out

in

PeP

EfficiencyEfficiency or Work WorkP Pt

t

0

i

PeP

0 i

PPower outePower in P

Efficiency is the ratio Efficiency is the ratio of the power output of the power output to the power input.to the power input.

Example 2.Example 2. A12A12--hphp winch motor lifts a winch motor lifts a 900900--lblb loadload

Po = 6270 ftlb/s

A simple A simple machinemachine

ss inin

ss outout

WW

FF inin = ?= ?

WW out

in

PeP

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

0

i

PeP

to a height of to a height of 8 ft8 ft. What is . What is the output power in the output power in ftftlb/slb/s if if the winch is 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 ft/s):(1 hp = 550 ft/s): 550 ft lb/s(11.4 hp) 6600 ft lb/s1 hpoP

0 iP eP

Ex. 2 (cont.)Ex. 2 (cont.) A12A12--hphp winch motor lifts a winch motor lifts a 900 lb900 lb loadload

A simple A simple machinemachine

ss inin

ss outout

WW

FF inin = ?= ?

WW out

in

PeP

EfficiencyEfficiencyWe just found that We just found that PPoo = 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 outPt t

(900 lb)(8 ft)6270

o o

o

F stP

Now we solve for Now we solve for t t ::

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

Actual Mechanical Actual Mechanical AdvantageAdvantage A simple A simple

machinemachiness inin

ss outout

WW

FF inin = ?= ?

WW

Actual Actual Mechanical Mechanical AdvantageAdvantageFFoutout

MMAA

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

oA

i

Foutput forceMinput 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 mechanical the actual mechanical advantage is:advantage is:

80 N40 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 frictionInput work = output work + work against friction

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

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

or o ii i o o

i o

F sF s F ss s

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

Ideal Mechanical Ideal Mechanical AdvantageAdvantage A simple A simple

machinemachiness inin

ss outout

WW

FF inin = ?= ?

WW

Ideal Ideal Mechanical Mechanical AdvantageAdvantageFFoutout

MMII

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

iA

o

sin distanceMout distance s

For example, if anFor example, if an inputinput force force moves a distance of moves a distance of 6 m6 m while while the the outputoutput force moves force moves 2 m2 m, the , the ideal mechanical advantage is:ideal mechanical advantage is:

6 m2 m3.0

I

I

M

M

6 m6 m

2 m2 m

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

A A simple simple machinmachin

eeSS in in = 8 m= 8 m

SS outout = 2 m= 2 m

WW

FF inin = 80 N= 80 N

WW ee = 100%= 100%

FFoutout == 400 N400 N

IDEAL EXAMPLE:IDEAL EXAMPLE:

8 m 42 m

iI

o

sMs

80 N 420 N

oA

i

FMF

1.0A

i

MeM

Efficiency for an Actual EngineEfficiency for an Actual EngineThe actual efficiency is always less than the The actual efficiency is always less than the ideal efficiency because friction always exits. ideal efficiency because friction always exits. The efficiency is still equal to the ratio The efficiency is still equal to the ratio MMAA /M/MII ..

A

i

MeM

The efficiency of any engine is given by:

In our previous example, the ideal mechanical advantage was equal to 4. If the engine was only 50% efficient, the actual mechanical advantage would be 0.5(4) or 2. Then 160 N (instead of 80 N) would be needed to lift the 400-N weight.

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

The LeverThe Lever

The input torque The input torque FFii rrii is equal is equal to the output torque to the output torque FFoo rroo ..

i i o oF r F r

The actual mechanical advantage is, therefore:

o iA

i o

F rMF r

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

FFoutout

rroutout rrinin

FulcrumFulcrum

The LeverThe LeverFriction is negligible Friction is negligible so that so that WWoutout = W= Winin ::

or oo

ii

io

i

Fs

sF

F F ss

The ideal MI is: and o iI I A

i o

F rM M MF r

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

FFinin

FFoutout rroutout

ssoutout

ssinin

rrinin

Example 3.Example 3. A A 11--mm metal lever is used to lift a metal lever is used to lift a 800800--N N rock. What force is required at the left end rock. What force is required at the left end if the fulcrum is placed if the fulcrum is placed 20 cm20 cm from the rock?from the rock?

rriirr22

800 N800 N

F = ?F = ?

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

FFoo = = 700 N; 700 N; rr22 = = 20 cm20 cm

rr11 = = 100 cm 100 cm -- 20 cm = 80 cm20 cm = 80 cm

3. To find 3. To find FFii we recall the definition of Mwe recall the definition of MII ::80 cm and 4 ;20 cm

iI I

o

rM Mr

For lever:For lever: MMAA = M= MII

4oA

i

FMF

Thus,Thus, 800 N 200 N4iF andand

Other Examples of LeversOther Examples of Levers

Wheel and Axel:Wheel and Axel:RR

rrFFii

FFoo

Wheel and AxelWheel and Axel

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

For Wheel and Axel:

o iA

i o

F rMF r

For example, if For example, if R = 30 cmR = 30 cm and and r = 10 cmr = 10 cm, an , an input force of only input force of only 100 N100 N will lift a will lift a 300300--NN weight!weight!

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

Single Fixed PulleysSingle Fixed PulleysSingle fixed pulleys serve only to change the Single fixed pulleys serve only to change the direction of the input force. See examples:direction of the input force. See examples:

W

FFininFFoutout

FFinin

FFoutout

FFin in = = FFoutout

Single Moveable PulleySingle Moveable Pulley

80 N80 N

FFinin

A freeA free--body diagram shows an body diagram shows an actualactual mechanical mechanical advantageadvantage of of MMAA == 22 for a single moveable pulley.for a single moveable pulley.

2 or 2oin out A

i

FF F MF

FFinin + F+ Fin in = = FFoutout

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

sMs

Block and Tackle Block and Tackle ArrangementArrangement

W FFoo

FFii

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

FFoo

FFiiFFii 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 FFMF

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

The Belt DriveThe Belt Drive

Belt Belt DriveDrive

FFoo

FFii

oA

i

output torqueMinput 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 MFr

o oI

i i

r DMr D

Belt Drive:rrii

rroo

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

o iI

i o

DMD

Belt Drive:

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

Belt Belt DriveDrive

DDoo

DDii

Speed ratio:Speed ratio: i

o

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

MMII = 4= 4

FFoo

FF rrii

rroo

To find DTo find Doo we use the fact thatwe use the fact that

4; 4oI o i

i

DM D DD

DD oo = 4(12 cm) = = 4(12 cm) = 48 cm48 cm

Now, Now, ii = = FFii rrii and and rrii = = DDii /2. /2. Belt tension is Belt tension is FFii and and rrii is is equal to equal to ½½DDii = 0.06 m.= 0.06 m.

200 N mi i iF r

200 N m 0.06

3 Nm

3 30iF

oI

i

DMD

GearsGears

o oI

i i

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

Example 5.Example 5. The driving gear on a bicycle has The driving gear on a bicycle has 4040 teeth and the wheel gear has only teeth and the wheel gear has only 2020 teeth. What teeth. What is the mechanical advantage? If the driving gear is the mechanical advantage? If the driving gear makes makes 60 rev/min60 rev/min, what is the rotational speed of , what is the rotational speed of the rear wheel? the rear wheel?

22 ; 0.544

oI I

i

NM MN

Remember that the Remember that the angular speed ratio is angular speed ratio is

opposite to the gear ratio.opposite to the gear ratio.1; 2

o i iI

i o o

NMN

oo = 2= 2

2(60 rpm)2(60 rpm)

Output angular speed:

= 120 rpm

Output angular speed:

= 120 rpm

Ni = 40No = 20

The Inclined PlaneThe Inclined Plane

FFii

FFoo = W= W

ssoo

ssii

The Inclined PlaneThe Inclined Plane

Ideal Mechanical Ideal Mechanical AdvantageAdvantage

iI

o

sslopeMheight s

Actual Advantage: Ai

WMF

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 of An inclined plane has a slope of 8 m8 m and a height of and a height of 2 m2 m. What is the ideal . What is the ideal mechanmechan-- icalical advantage and what is the necessary input advantage and what is the necessary input force needed to push a force needed to push a 400400--N N weight up the weight up the incline? The efficiency is incline? The efficiency is 60 percent60 percent..

FFii

FFoo = 400 N= 400 N

2 m2 m

SSii = = 8 m8 m

8 m ;2 m

4 iI I

o

sMs

M

; (0.60)(4)AA I

I

Me M eMM

2.4 oA

i

FMF

400 N2.4 2.4

oi

FF Fi = 167 NFi = 167 N

The Screw JackThe Screw Jack

p

FoFiR

Screw Jack

2I

RMp

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

Input distance: Input distance: ssii = 2= 2R R Output distance: Output distance: ssoo = p= p

2i

Io

Screw Jacks RMs p

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

Summary for Simple MachinesSummary for Simple Machines

0 i

PPower outePower in P

Efficiency is the ratio Efficiency is the ratio of the power output of the power output to the power input.to the power input.

EfficiencyEfficiency ee is defined is defined as the ratio of work as the ratio of work

output to work input.output to work input.

out out

in in

Work outputeWork inputF seF s

SummarySummary

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

ratio of ratio of ssin in to to ssoutout ..

iA

o

sin distanceMout distance s

The The actual mechanical actual mechanical advantageadvantage,, MMAA , is the , is the

ratio of ratio of FFoo to to FFii ..

oA

i

Foutput forceMinput force F

A simple A simple machinemachine

ss inin

ss outout

WW

FF inin = ?= ?

WW out

in

PeP

EfficiencyEfficiency

Summary (Cont.)Summary (Cont.)

The actual mechanical advantage for a lever:

o iA

i o

F rMF r

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

For Wheel and axel:

o iA

i o

F rMF r

Summary (Cont.)Summary (Cont.)

oA

i

output torqueMinput torque

o oI

i i

r DMr D

Belt Drive:

o iI

i o

DMD

Belt Drive:MMII = 4= 4

FFoo

FF rrii

rroo

Belt Belt DriveDrive

SummarySummary

o oI

i i

D NMD N

Gears:

FFii

FFoo = W= W

ssoo

ssii

The Inclined The Inclined PlanePlane

Ideal Mechanical Ideal Mechanical AdvantageAdvantage

iI

o

sslopeMheight s

Actual Advantage: Ai

WMF

NiNo

Summary (Cont.)Summary (Cont.)

p

FoFiR

Screw Jack

2I

RMp

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

Input distance: Input distance: ssii = 2= 2R R Output distance: Output distance: ssoo = p= p

2i

Io

Screw Jacks RMs p

CONCLUSION: Chapter 12 CONCLUSION: Chapter 12 Simple MachinesSimple Machines