Simple Machines Math (Mechanical Advantage, Efficiency, and Energy)

Simple Simple MachinesMachines

MathMath(Mechanical Advantage, (Mechanical Advantage, Efficiency, and Energy)Efficiency, and Energy)

What is an instrument that

makes work easier called?

What is an instrument that

makes work easier called?

What are the two types of work

involved in using a machine?

What is an instrument that makes work easier

called?

Describe the two types of

work involved in using a hammer.


work involved in using scissors.


work involved in using a rolling pin.

Can a machine increase the work you

put into it?

Can a machine increase the work you

put into it?

Mechanical Mechanical AdvantageAdvantage

Mechanical Advantage – a quantity that measures how much a machine multiplies force or distance.


The mechanical advantage tells you how much the force will increase by using a particular machine. The more times a machine multiplies the input or effort force, the better the machine is.


When the mechanical advantage is greater than 1, the machine increases or multiplies the force you apply.


Machines do not increase the work you put into them.

The work that goes into a machine can never be greater than the work that comes out of the machine.

Mechanical Mechanical Advantage EquationAdvantage Equation

mechanical advantage = output force = input distance input force output distance

M.A. = Fo = di

Fi do

mechanical advantage = resistance force = effort distance effort force resistance distance

M.A. = FR = dE

FE dR

The resistance force can be just the weight of the object you are trying to move. (Weight is a force of gravity.)

Mechanical advantage has no units.

M.A. Problem 1M.A. Problem 1What is the mechanical advantage of a crowbar that allows you to put 25 newtons of force into lifting a 250 newton crate?

M.A. Problem 1M.A. Problem 1What is the mechanical advantage of a crowbar that allows you to put 25 newtons of force into lifting a 250 newton crate?

M.A. = Fo Fi

M.A. = 250 N = 10 25 N

M.A. = FR FE

M.A. = 250 N = 10 25 N

How many times does the crowbar multiply the force that wasput into it? (Hint: Look at the answer.)

M.A. Problem 2M.A. Problem 2What is the mechanical advantage of ramp that is 10 m long and 3 m high?

M.A. Problem 2M.A. Problem 2What is the mechanical advantage of ramp that is 10 m long and 3 m high?

M.A. = di do

M.A. = 10 m = 3.3 3 m

M.A. = dE dR

M.A. = 10 m = 3.3 3 m

M.A. Problem 3M.A. Problem 3A pulley system has a mechanical advantage of 10. a. If a mover uses this pulley to lift a piano with a weigh of 1450 N a distance of 4 m, how much force must the mover use?b. How far will the mover pull the rope?

M.A. Problem 3M.A. Problem 3

a. M.A. = Fo Fi

10 = 1450 N Fi

Fi = 1450 N 10

Fi = 145 N

A pulley system has a mechanical advantage of 10. a. If a mover uses this pulley to lift a piano with a weigh of 1450 N a distance of 4 m, how much force must the mover use?b. How far will the mover pull the rope?

b. M.A. = di do

10 = di 4 m

(10)(4m) = di

di = 40 m

Can the work that comes out of a

machine be greater than the work that goes

into the machine?

Can the work that comes out of a machine be greater than the work that goes into the

machine?

Are any actual actual machinesmachines

100% efficient?

Are any actual machines 100% efficient?

There are no machines that are 100% efficient. Every machine deals with friction…some more than others. Friction is a force that opposes motion.

Ideal MachinesIdeal Machines

Ideal Machines are 100% efficient.

Ideal Machines do not exist.

What form of energy does friction produce?

What form of energy does friction produce?

heat

Why? Friction opposes motion.

EfficiencyEfficiencyThe efficiency of a machine is defined as the

ratio of the output work to the input work.

efficiency = work output x 100% work input

eff = Wo x 100% Wi

EfficiencyEfficiencyIn an ideal machine…

- work output equals work input. - the efficiency is 100%.

In real machines… - the efficiency is less than 100%. - work output is less than work input. - loss due to friction and heat.

Efficiency Problem 1Efficiency Problem 1 A man uses 419 J of work in removing a

nail from a piece of wood with a hammer. The hammer has a work output of 305 J. What is the efficiency of the hammer?

Efficiency Problem 1Efficiency Problem 1 A man uses 419 J of work in removing a nail

from a piece of wood with a hammer. The hammer has a work output of 305 J. What is the efficiency of the hammer?

eff = Wo x 100% Wi

eff = 305 J x 100 419 J

eff = 72.8 %

Efficiency Problem 2Efficiency Problem 2 John uses 39 J of energy to

move four boxes with the handcart. The work output from the handcart is 32.4 J. What is the efficiency of the handcart?

Efficiency Problem 2Efficiency Problem 2 John uses 39 J of energy to move four

boxes with the handcart. The work output from the handcart is 32.4 J. What is the efficiency of the handcart?

eff = Wo x 100% Wi

eff = 32.4 J x 100 39 J

eff = 83 %

Moment ProblemsMoment Problems

Moment = length mass

momentresistance = momenteffort

(length mass)resistance = (length mass)effort


45 g 22.3 g

resistance arm5 m

effort arm? m

Solve for the missing quantity.



(45 g 5 m) = (22.3 g X)

X = 10.09 m

45 g 22.3 g

resistance arm5 m

effort arm? m


38 g 13.5 g

resistance arm3 m

effort arm? m

Solve for the missing quantity.


38 g 13.5 g

resistance arm3 m

effort arm? m


(38 g 3 m) = (13.5 g X)

X = 8.4 m

EnergyEnergy

Energy is the ability to do work.

Potential Energy – energy at rest due to position, composition, or compression.

Kinetic Energy – energy of motion

Potential EnergyPotential Energypotential energy = mass free fall acceleration height

PE = m g h

J = kg m/sec2 m

1 joule = 1 kg m2

sec 2

g = 9.8 m/sec2

Potential Energy Problem Potential Energy Problem 11

A rock climber climbs 63 m to the top of a cliff. If the rock climber has

a mass of 85 kg, what is the potential energy of the climber?


A rock climber climbs 63 m to the top of a cliff. If the rock climber has a mass of 85 kg, what is the potential energy of the climber?

PE = m g h

PE = (85 kg)(9.8 m/sec2)( 63 m)

PE = 52,479 J


A 1.8 kg book sits on top of a 2.8 m shelf. What is the potential energy of the book?


A 1.8 kg book sits on top of a 2.8 m shelf. What is the potential energy of

the book?

PE = m g h

PE = (1.8 kg)(9.8 m/sec2)( 2.8 m)

PE = 49.39 J

Kinetic EnergyKinetic EnergyKinetic energy = ½ mass speed squared

KE = ½ m v2

J = ½ kg m2/sec2

1 joule = 1 kg m2

sec 2

Kinetic Energy Problem 1Kinetic Energy Problem 1

What is the kinetic energy of a 35 kg dog running at 2.8 m/sec?


What is the kinetic energy of a 35 kg dog running at 2.8 m/sec?

KE = ½ m v2

KE = ½ 35 kg (2.8 m/sec)2

KE = 137.2 J


What is the kinetic energy of a 1635 kg car traveling at 29 m/sec?


What is the kinetic energy of a 1635 kg car traveling at 29 m/sec?

KE = ½ m v2

KE = ½ 1635 kg (29 m/sec)2

KE = 687,517.5 J

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Simple Machines Math (Mechanical Advantage, Efficiency, and Energy)