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

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Simple MachinesMath(Mechanical Advantage,
Efficiency, and Energy)
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What is an instrument that makes work easier
called?
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What is an instrument that makes work easier
called?
a machine
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What are the two types of work involved in using
a machine?
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What is an instrument that makes work easier
called?
Work that goes into a machine (work input)
and work that come out of a machine (work
output).
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Describe the two types of work involved in using
a hammer.
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Describe the two types of work involved in using
scissors.
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Describe the two types of work involved in using
a rolling pin.
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Can a machine increase the work you put into it?
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Can a machine increase the work you put into it?
No, machines can only change the factors that
determine work.
work force x distance
Either the force will be multiplied or the
distance will be multiplied but never both at the
same time.
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Mechanical Advantage

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

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Mechanical Advantage

• 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.

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Mechanical Advantage

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

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Mechanical Advantage

• 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.

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Mechanical Advantage 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.

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M.A. Problem 1

• What is the mechanical advantage of a crowbar
  that allows you to put 25 newtons of force into
  lifting a 250 newton crate?

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M.A. Problem 1

• What 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.)
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M.A. Problem 2

• What is the mechanical advantage of ramp that is
  10 m long and 3 m high?

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M.A. Problem 2

• What 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
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M.A. Problem 3

• 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?

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M.A. Problem 3

• 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?

a. M.A. Fo Fi 10
1450 N Fi Fi 1450 N
10 Fi 145 N
b. M.A. di do
10 di 4 m (10)(4m)
di di 40 m
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Can the work that comes out of a machine be
greater than the work that goes into the machine?
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Can the work that comes out of a machine be
greater than the work that goes into the machine?
No, because every machine has some type of
friction.
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Are any actual machines 100 efficient?
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Are any actual machines 100 efficient?
There are no machines that are 100 efficient.
Every machine deals with frictionsome more than
others. Friction is a force that opposes motion.

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Ideal Machines

• Ideal Machines are 100 efficient.
• Ideal Machines do not exist.

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What form of energy does friction produce?
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What form of energy does friction produce?
heat
Why? Friction opposes motion.
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Efficiency

• The 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

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Efficiency

• In 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.

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Efficiency 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?

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Efficiency 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
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Efficiency 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?

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Efficiency 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
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Moment Problems
Moment length ? massmomentresistance
momenteffort(length ? mass)resistance (length
?mass)effort
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Moment Problems
resistance arm5 m
effort arm? m
45 g
22.3 g
Solve for the missing quantity.
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Moment Problems
resistance arm5 m
effort arm? m
(length ? mass)resistance (length
?mass)effort(45 g ? 5 m) (22.3 g ? X)X
10.09 m
45 g
22.3 g
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Moment Problems
resistance arm3 m
effort arm? m
38 g
13.5 g
Solve for the missing quantity.
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Moment Problems
resistance arm3 m
effort arm? m
38 g
13.5 g
(length ? mass)resistance (length
?mass)effort(38 g ? 3 m) (13.5 g ? X)X
8.4 m
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Energy
Energy is the ability to do work.Potential
Energy energy at rest due to position,
composition, or compression.Kinetic Energy
energy of motion
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Potential Energy
potential energy mass ? free fall acceleration
? height PE m ? g ? hJ kg ? m/sec2 ? m1
joule 1 kg ? m2
sec 2g 9.8 m/sec2
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Potential Energy Problem 1
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?
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Potential Energy Problem 1
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
? hPE (85 kg)(9.8 m/sec2)( 63 m)PE 52,479
J
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Potential Energy Problem 2
A 1.8 kg book sits on top of a 2.8 m shelf. What
is the potential energy of the book?
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Potential Energy Problem 2
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
? hPE (1.8 kg)(9.8 m/sec2)( 2.8 m)PE
49.39 J
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Kinetic Energy
Kinetic energy ½ ? mass ? speed squared KE
½ ? m ? v2J ½ ? kg ? m2/sec2 1 joule 1
kg ? m2 sec 2
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Kinetic Energy Problem 1
What is the kinetic energy of a 35 kg dog running
at 2.8 m/sec?
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Kinetic Energy Problem 1
What is the kinetic energy of a 35 kg dog running
at 2.8 m/sec? KE ½ ? m ? v2KE ½ ? 35 kg ?
(2.8 m/sec)2 KE 137.2 J
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Kinetic Energy Problem 2
What is the kinetic energy of a 1635 kg car
traveling at 29 m/sec?
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Kinetic Energy Problem 2
What is the kinetic energy of a 1635 kg car
traveling at 29 m/sec? KE ½ ? m ? v2KE ½ ?
1635 kg ? (29 m/sec)2 KE 687,517.5 J