PHY 126 Lecture Notes Chapter 10

PHY 126 Lecture Notes Chapter 10

1 | Susan S. Gorelick Most recent revision

Chapter 10 Simple Machines

OBJECTIVES

Define a machine

Examine energy transfer in machine to determine Mechanical Advantage and Energy Efficiency

KEY WORDS: Simple and complex machines, Effort and resistance forces, Mechanical Advantage,

Energy efficiency; Law of Simple machines and Conservation of Work or Energy; Compound machine;

Ideal mechanical advantage vs. actual mechanical advantage.

MACHINES AND ENERGY TRANSFER

A machine is an object, device, or system that is used to transfer energy from one place to another and

allows work to be done that may not be able to be done otherwise.

A simple machine is a device with few moving parts that allow the user to convert an applied force to

some type of useful work.

There are six types of simple machines: (we will also examine gear)

Lever

Wheel and axle

Pulley

Ramp (or Inclined plane)

Screw

Wedge

Simple machines allow us to:

Multiply force: wrench, hammer

Multiply speed: gears

Change direction: pulley, gear

A complex machine is just some type of combination of simple machines. A car, for example, is an

example of a complex machine. Our human body can be considered a complex machine!

There are two forces to concern ourselves with when it comes to machines: effort and resistance.

The effort force is the force that is applied to the machine.

The resistance force is the force that the machine must overcome to do work.

Example – A person applies 20 lb of force to a jack. This is the effort force. The jack produces 600 lb of

force to lift the object. This is the resistance force.



The mechanical advantage of a simple machine is defined as the ratio of the resistance force to the

effort force.

EXAMPLE 1: Find the MA of a jack that requires 20 lb of applied force to lift a 600 lb object

GIVEN FIND FORMULA

Resistance force = 600 lb

Effort force = 20 lb

MA =?

𝑴𝑨 =𝟑𝟎

𝟏

MA =resistance force

effort force

MA =600 lb

20 lb =

𝟑𝟎𝟏

The mechanical advantage of the jack is 30 to 1, which

means the jack can lift 30 lb for every lb of force

applied to the jack.

Due to outside forces, such as friction, energy can be lost and output can be lessened.

𝑴𝑨 =𝒓𝒆𝒔𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒇𝒐𝒓𝒄𝒆

𝒆𝒇𝒇𝒐𝒓𝒕 𝒇𝒐𝒓𝒄𝒆

MA = the mechanical advantage of the machine

MA is unit-less because it is a ratio.

The higher the mechanical advantage, the greater the output of the

machine.

% 𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒄𝒚 =𝒘𝒐𝒓𝒌 𝒐𝒖𝒕𝒑𝒖𝒕

𝒘𝒐𝒓𝒌 𝒊𝒏𝒑𝒖𝒕× 𝟏𝟎𝟎%

The more output you are able to achieve, the more efficient the machine

The more friction acting on the system, the less efficient the machine will be



THE LAW OF CONSERVATION OF ENERGY (or WORK) LAW OF SIMPLE MACHINES

When a machine increases force or speed, there is always a price to be paid because work must be

conserved. Hence, it remains constant throughout the system. Recalling W = F*s, so even when we gain a

mechanical advantage using a simple machine, the total amount of work done remains the same (is

conserved).

The law of simple machines states that work must be conserved and remain constant throughout the

system.

THE LEVER

The lever is a simple machine used to multiply force.

The lever consists of a bar that turns on a pivot, known as the fulcrum. resistance force

effort force

effort arm fulcrum resistance arm

The effort arm is the distance from the effort force to the fulcrum.

The resistance arm is the distance from the resistance force to the fulcrum

Due to conservation of work, the following can be said about levers:

𝑭𝑹 ∙ 𝒔𝑹 = 𝑭𝑬 ∙ 𝒔𝑬

FR = resistance force

sR = length of resistance arm

FE = effort force

sE = length of effort arm

𝒓𝒆𝒔𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒇𝒐𝒓𝒄𝒆 × 𝒓𝒆𝒔𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆 = 𝒆𝒇𝒇𝒐𝒓𝒕 𝒇𝒐𝒓𝒄𝒆 × 𝒆𝒇𝒇𝒐𝒓𝒕 𝒅𝒊𝒔𝒕𝒂𝒏𝒄𝒆



The mechanical advantage of a lever can be found by using the resistance and effort arms by the

following formula:

Before we can put this into action we must classify the different types of levers we use:

First Class

A seesaw, your neck, and a jack are examples of this type of lever.

Second Class

A wheel barrel and the ball of your foot are examples of this type of lever.

FR

FE

sR FR sE

FE sE

sR

𝑴𝑨𝒍𝒆𝒗𝒆𝒓 =𝒆𝒇𝒇𝒐𝒓𝒕 𝒂𝒓𝒎

𝒓𝒆𝒔𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒂𝒓𝒎=

𝒔𝑬

𝒔𝑹

MAlever = the mechanical advantage of the lever

sE = effort arm

sR = resistance arm



Third Class

A hockey stick, fishing pole, and your forearm are examples of this type of lever.

EXAMPLE 1: A wheel barrel is 1.20 m long and has a 900 N load 40.0 cm from the axle.

A) What force is needed to lift the wheel barrel? B) What is the MA?

GIVEN FIND FORMULA

SE = 1.20 m

FR = 900 N

SR = 0.40 m

Second-Class Lever.

Why?

FE =?

MA = ?

FR ∙ sR = FE ∙ sE

(900 N) ∙ (0.40 m) = FE ∙ (1.20 m)

𝑭𝑬 = 𝟑𝟎𝟎 𝑵

MAlever =sE

sR

MAlever =1.20 m

0.40 m

𝑴𝑨𝒍𝒆𝒗𝒆𝒓 =𝟑

𝟏

EXAMPLE 2: Two children are sitting on a see-saw. One child weighs 75 lb and is seated 3 feet from

the fulcrum. The other child weighs 60 lb. How far from the fulcrum must the second child sit so that the

see-saw remains balanced?

GIVEN FIND FORMULA

FR = 75 lb

SR = 3 ft

FE = 60 lb

First Class Lever. Why?

SE

FR ∙ sR = FE ∙ sE

(75 lb) ∙ (3 ft) = (60 lb) ∙ SE

𝑺𝑬 = 𝟑. 𝟕𝟓 𝒇𝒕

sE FE

sR

FR

The second child must sit 3.75 ft

away from the fulcrum for the

see-saw to remain balanced.

300 N of force is required

to lift the wheel barrel.

The mechanical advantage

is 3 to 1, which indicates

that the barrel can produce

a resistance force three

times that of the effort force



THE WHEEL-AND-AXLE

Where FR = resistance force

rR = radius of resistance force

FE = effort force

RE = radius of effort force

MAwheel-and-axle = 𝒓𝒂𝒅𝒊𝒖𝒔 𝒐𝒇 𝒆𝒇𝒇𝒐𝒓𝒕 𝒇𝒐𝒓𝒄𝒆

𝒓𝒂𝒅𝒊𝒖𝒔 𝒐𝒇 𝒓𝒆𝒔𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒇𝒐𝒓𝒄𝒆 =

𝒓𝑬

𝒓𝑹

EXAMPLE: (Example 1 on textbook p. 274)

A winch has a handle that turns in a radius of 30.0 cm. The radius of the drum or axle is 10.0 cm. Find

the force required to lift a bucket weighing 500 N (Fig. 10.12)

Given Find Formula

FR = 500 N FE = ? FR*rR = FE* rE

rE = 30.0 cm Rearrange for FE:

rR = 10.0 cm FE =

FR∗rR

𝑟𝐸

FE =

(500N)(10.0cm)

30.0cm

= 167 N

FR*rR = FE*rE



THE PULLEY


The pulley system in Fig. 10.16 is used to raise a 650-lb object 25 ft. What is the mechanical advantage?

What force is exerted?

Given Find Formula

MApulley = 5 FE = ? MApulley = FR/FE

because there are 5

strands holding the

resistance

Rearrange for FE:

FE = FR/MApulley

FR = 650 lb FE = 650 lb/5

= 130 lb

FR*sR = FE*sE

where s refers to the distance

moved

𝐹𝑅

𝐹𝐸 =

𝑠𝐸

𝑠𝑅 = MA pulley

MApulley = number of strands

holding the resistance



RAMP OR INCLINED PLANE


Find the length of the shortest ramp that can be used to push a 600-lb resistance onto a platform 3.5-ft

high by exerting a force of 72.0 lb.

Given Find Formula

FR = 600 lb length = ? FR*height = FE*length

FE = 72 lb Rearrange for length:

height = 3.5 ft length = (FR*height)/FE

length =

(600 lb)(3.5ft)

72.0 lb

= 29.2 ft

THE SCREW

A screw is an inclined plane wrapped around a cylinder.

The jackscrew, wood screw, and auger are examples.

The rise distance a beam rises or the distance the wood screw advances into a piece of wood in

one revolution is called the pitch or the lead of the screw.

From the law of machines, FR*sR = FE*sE

o For advancing a screw with a screwdriver,

sR = pitch or lead of screw

sE circumference of the handle of the screwdriver

Law of Machines:

FR*sR = FE*sE

FR*height of ramp = FE*length of ramp

𝑭𝑹

𝑭𝑬 =

𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒓𝒂𝒎𝒑

𝒉𝒆𝒊𝒈𝒉𝒕 𝒐𝒇 𝒓𝒂𝒎𝒑 =

MA inclined plane or ramp

𝑭𝑹

𝑭𝑬 =

𝟐𝝅𝒓

𝒑𝒊𝒕𝒄𝒉 𝒐𝒓 𝒍𝒆𝒂𝒅

where r is the radius of

the handle of the

screwdriver




A 19,400-N weight is raised using a jackscrew having a pitch of 5.00 mm and a handle length of 255 mm.

What force must be applied?

Given Find Formula

pitch = 5.00 mm FE = ? FR*pitch = FE*2πr

r = 255 mm Rearrange for FE:

FR = 19,400 N FE =

FR∗pitch

2πr

FE =

(19,400 𝑁)(5.00 𝑚𝑚)

2π(255 mm)

= 60.5 N

THE WEDGE

A wedge is an inclined plane in which the plane is moved

instead of the resistance.

Finding the mechanical advantage of a wedge is not practical

because of the large amount of friction.

A narrow wedge is easier to drive than a thick wedge.

Therefore, the mechanical advantage depends on the ratio of its

length to its thickness.



THE GEAR

Compound Machines

A compound machine is a combination of simple machines.

In most compound machines, the total mechanical advantage is the product of the mechanical

advantage of each simple machines.

EXAMPLE: (Example on textbook p. 287)

A crate weighing 9500 N is pulled up the inclined plane using the pulley system shown in Fig. 10.23.

(a) Find the mechanical advantage of the total system.

(b) What effort force (FE) is needed?

Given Find Formula

(a) length = 10.5 m MA inclined plane = ? MA inclined plane = length of plane

height of plane =

10.5 m

1.50 m = 7.00

height = 1.50 m MA total system =?

MA pulley = 5 MA total system = (MA inclined plane)(MA pulley)

(the number of supporting

strands) = (7.00)(5) = 35.0

(b)MA compound machine = 35.0 FE = ? MA compound machine = 𝐹𝑅

𝐹𝐸

Rearrange for FE

FR = 9500 N FE = FR/MA compound machine =9,500 N

35.0 = 271 N

MA compound machines = (MA1)(MA2)(MA3) …..



The Effect of Friction on Simple Machines

So far the law of simple machines has been stated in terms of ideal mechanical advantage

(IMA), in which we have 100% efficiency.

In reality, energy is lost in every machine through heat to overcome friction.

This lost energy decreases the efficiency of the machine; output work is always less than input

work.

This lost energy is heat energy, which results in machine wear or even burning out of certain parts

of the machine.

In general, the actual mechanical advantage is found by:

EXAMPLE: In Example 1 of Section 10.5, the inclined plane has an IMA of 4 to 1. Actually, it takes

545 N of effort to move the 1500-N box up the ramp. Therefore, the AMA is

AMA = 𝐹𝑅

𝐹𝐸 =

1500 𝑁

545 𝑁 = 2.75

AMA = 𝑭𝑹

𝑭𝑬 =

𝒓𝒆𝒔𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒇𝒐𝒓𝒄𝒆

𝒆𝒇𝒇𝒐𝒓𝒕 𝒇𝒐𝒓𝒄𝒆

PHY 126 Lecture Notes Chapter 10

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