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### Chapter 12 – Simple Machines

A PowerPoint Presentation by

Paul E. Tippens, Professor of Physics

Southern Polytechnic State University

© 2007

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

Photo Vol. 1 PhotoDisk/Getty

Objectives: After completing this module, you should be able to:

- Describe a simple machine in general terms and apply the concepts of efficiency, energy conservation, work, and power.
- Distinguish by definition and example between the concepts of the ideal and actual mechanical advantages.
- Describe and apply formulas for the mechanical advantage and efficiency of the following devices: (a) levers, (b) inclined planes, (c) wedges, (d) gears, (e) pulley systems, (f) wheel and axel, (g) screw jacks, and (h) the belt drive.

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

Fin

A simple machine

sin

Win= Finsin

sout

Fout

W

Wout= Foutsout

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

Fin

Fin

A simple machine

sin

Win= Finsin

sout

Fout

Fout

W

W

Wout= Foutsout

A Simple Machine (Cont.)Input work = output work + work against friction

Efficiency eis defined as the ratio of work output to work input.

A simple machine

sin

Efficiency

sout

W

W

The advantage is a reduced input force, but it is at the expense of distance. The input force must move a greater distance.

sin = 5.0m

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

The efficiency is 80% or e = 0.80, therefore

A simple machine

sin

Efficiency

sout

W

W

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

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

A simple machine

sin

First we must find the power output, Po:

Efficiency

sout

W

W

Po = 6270 ftlb/s

Example 2.A12-hp winch motor lifts a 900-lb loadto a height of 8 ft. What is the output power in ftlb/s if the winch is 95% efficient?

Po= (0.95)(12 hp) = 11.4 hp

(1 hp = 550 ft/s):

A simple machine

sin

Efficiency

sout

W

W

Ex. 2 (cont.)A12-hp winch motor lifts a 900 lb loadto a height of 8 ft. How much time is required if the winch is 95% efficient?

We just found that Po = 6270 W

Now we solve for t :

Time required: t = 1.15 s

A simple machine

sin

Actual Mechanical Advantage

Fout

sout

W

W

MA

40 N

80 N

Actual Mechanical AdvantageThe actual mechanical advantage, MA, is the ratio of Fo to Fi.

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

Conservation of energy demands that:

Input work = output work + work against friction

Anidealor perfect machine is 100% efficient and (Work)f = 0, so that

An Ideal MachineThe ratio si/sois the ideal mechanical advantage.

6 m

A simple machine

sin

Ideal Mechanical Advantage

2 m

Fout

sout

W

W

MI

Ideal Mechanical AdvantageThe ideal mechanical advantage,MI, is the ratio of sin to sout.

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

Fin = 80 N

A simple machine

Sin = 8 m

Fout=400 N

Sout= 2 m

W

W

e = 100%

Efficiency for an Ideal EngineFor 100% efficiency MA = MI. In other words, in the absence of friction, the machine IS an ideal machine and e = 1.

The efficiency of any engine is given by:

Efficiency for an Actual EngineThe actual efficiency is always less than the ideal efficiency because friction always exits. The efficiency is still equal to the ratio MA/MI.

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.

Friction is negligible so that Wout = Win:

Fout

rout

sin

q

rin

q

sout

Fin

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

The idealMIis:

The Leverri

r2

Fo = 700 N; r2 = 20 cm

F = ?

r1 = 100 cm - 20 cm = 80 cm

Thus,

and

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

1. Draw and label sketch:

2. List given info:

3. To find Fi we recall the definition of MI :

For lever:MA = MI

Application of Lever Principle:

With no friction MI = MAand

For Wheel and Axel:

R

Fi

r

Fo

Wheel and Axel

Wheel and Axel:For example, if R = 30 cm and r = 10 cm, an input force of only 100 N will lift a 300-N weight!

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

Fout

Fout

Fin

W

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

Fin = Fout

Fin

Fin

Fin + Fin = Fout

2 m

1 m

80 N

40 N + 40 N= 80 N

Fout

80 N

Note that the rope moves a distance of 2 m while the weight is lifted only 1 m.

Single Moveable PulleyA free-body diagram shows an actualmechanical advantage of MA = 2 for a single moveable pulley.

Fi

Fi

Fi

Fi

Fo

Fo

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

W

Block and Tackle ArrangementWe draw a free-body diagram:

ro

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

Belt Drive

ri

Fi

Belt Drive:

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

wo

wi

Belt Drive

Di

Belt Drive:

Speed ratio:

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

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

MI = 4

ro

To find Do we use the fact that

ri

Fi

Example 4.A 200 Nm torque is applied to an input pulley12 cm in diameter. (a) What should be the diameter of the output pulley to give an ideal mechanical advantage of 4? (b) What is the belt tension?

Do = 4(12 cm) = 48 cm

Now, ti = Firi and ri = Di/2. Belt tension is Fi and ri is equal to ½Di = 0.06 m.

No

Ni

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

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

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

Ni = 40

Example 5.The driving gear on a bicycle has 40 teeth and the wheel gear has only 20 teeth. What is the mechanical advantage? If the driving gear makes 60 rev/min, what is the rotational speed of the rear wheel?

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

Output angular speed:

wo = 120 rpm

wo = 2wi = 2(60 rpm)

The Inclined Plane

si

Fi

so

q

Fo = W

The Inclined PlaneBecause of friction, the actual mechanical advantage MAof an inclined plane is usually much less than the ideal mechanical advantage MI.

Fi

2 m

q

Fo = 400 N

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

Fi = 167 N

An application of the inclined plane:

Fo

Input distance: si = 2pR

Fi

R

Output distance: so = p

p

Screw Jack

The Screw JackDue to friction, the screw jack is an inefficient machine with an actual mechanical advantage significantly less than the ideal advantage.

Efficiency eis defined as the ratio of work output to work input.

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

Summary for Simple MachinesApplication of lever principle:

With no friction MI = MA

For Wheel and axel:

The actual mechanical advantage for a lever:

Summary (Cont.)An application of the inclined plane:

Fo

Input distance: si = 2pR

Fi

R

Output distance: so = p

p

Screw Jack

Summary (Cont.)
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