2.1d Mechanics Work, energy and power

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2.1d Mechanics Work, energy and power

• Breithaupt pages 148 to 159

April 14th, 2012
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AQA AS Specification
Lessons Topics
1 2 Work, energy and power W Fs cos ? P ?W / ?t P Fv
3 4 Conservation of energy Principle of conservation of energy, applied to examples involving gravitational potential energy, kinetic energy and work done against resistive forces. ?Ep mg?h Ek ½ mv2
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Work (W)

• Work is done when a force moves its point of
  application.
• work force x distance moved in the
  direction of the force
• W F s
• unit joule (J)
• work is a scalar quantity

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• If the direction of the force and the distance
  moved are not in the same direction

W F s cos ?
The point of application of force, F moves
distance s cos ? when the object moves through
the distance s.
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Question 1

• Calculate the work done when a force of 5
  kN moves through a distance of 30 cm
• work force x distance
• 5 kN x 30 cm
• 5000 N x 0.30 m
• work 1500 J

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Question 2

• Calculate the work done by a child of weight 300N
  who climbs up a set of stairs consisting of 12
  steps each of height 20cm.
• work force x distance
• the child must exert an upward force equal to its
  weight
• the distance moved upwards equals (12 x 20cm)
  2.4m
• work 300 N x 2.4 m
• work 720 J

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Question 3

• Calculate the work done by the wind on the yacht
  in the situation shown below

• W F s cos ?
• 800 N x 50 m x cos 30
• 40 000 x cos 30
• 40 000 x 0.8660
• work 34 600 J

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Complete
Answers
Force Distance Angle between F and s Work
400 N 5 km 0 2 MJ
200 µN 300 m 0 60 mJ
50 N 6 m 60 150 J
400 N 3 m 90 0 J
400 N
300 m
60
0 J
Note No work is done when the force and
distance are perpendicular to each other.
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Force-distance graphs

• The area under the curve is equal to the work
  done.

area work ½ F s
area work found by counting squares on the graph
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Question

• Calculate the work done by the brakes of a car if
  the force exerted by the brakes varies over the
  cars braking distance of 100 m as shown in the
  graph below.

• Work area under graph
• area A area B
• (½ x 1k x 50)
• (1k x 100)
• (25k) (100k)
• work 125 kJ

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Energy (E)

• Energy is needed to move objects, to change their
  shape or to warm them up.
• Work is a measurement of the energy required to
  do a particular task.
• work done energy change
• unit joule (J)

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Conservation of Energy

• The principle of the conservation of energy
  states that energy cannot be created or
  destroyed.
• Energy can change from one form to another.
• All forms of energy are scalar quantities

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Some examples of forms of energy

• Kinetic energy (KE)
• Energy due to a bodys motion.
• Potential energy (PE)
• Energy due to a bodys position
• Thermal energy
• Energy due to a bodys temperature.
• Chemical energy
• Energy associated with chemical reactions.

• Nuclear energy
• Energy associated with nuclear reactions.
• Electrical energy
• Energy associated with electric charges.
• Elastic energy
• Energy stored in an object when it is stretched
  or compressed.

All of the above forms of energy (and others) can
ultimately be considered to be variations of
kinetic or potential energy.
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Kinetic Energy (EK)

• Kinetic energy is the energy an object has
  because of its motion and mass.
• kinetic energy ½ x mass x (speed)2
• EK ½ m v2
• Note v speed NOT velocity.
• The direction of motion has no relevance to
  kinetic energy.

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Question 1

• Calculate the kinetic energy of a car of mass 800
  kg moving at 6 ms-1
• EK ½ m v2
• ½ x 800kg x (6ms-1)2
• ½ x 800 x 36
• 400 x 36
• kinetic energy 14 400 J

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Question 2

• Calculate the speed of a car of mass 1200kg if
  its kinetic energy is 15 000J
• EK ½ m v2
• 15 000J ½ x 1200kg x v2
• 15 000 600 x v2
• 15 000 600 v2
• 25 v2
• v ?25
• speed 5.0 ms-1

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Question 3

• Calculate the braking distance a car of mass 900
  kg travelling at an initial speed of 20 ms-1 if
  its brakes exert a constant force of 3 kN.
• k.e. of car ½ m v2
• ½ x 900kg x (20ms-1)2
• ½ x 900 x 400
• 450 x 400
• k.e. 180 000 J

• The work done by the brakes will be equal to this
  kinetic energy.
• W F s
• 180 000 J 3 kN x s
• 180 000 3000 x s
• s 180 000 / 3000
• braking distance 60 m

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Complete
Answers
Mass Speed Kinetic energy
400 g 4.0 ms-1 3.2 J
3000 kg 10 kms-1 60 mJ
8 kg 300 cms-1 36 J
50 mg 12 ms-1 3.6 mJ
3.2 J
1.5 x 1011 J
8 kg
12 ms-1
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Gravitational Potential Energy (gpe)

• Gravitational potential energy is the energy an
  object has because of its position in a
  gravitational field.
• change in g.p.e.
• mass x gravitational field strength x
  change in height
• ?EP m g ?h

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Question

• Calculate the change in g.p.e. when a mass of 200
  g is lifted upwards by 30 cm.
• (g 9.8 Nkg-1)
• ?EP m g ?h
• 200 g x 9.8 Nkg-1 x 30 cm
• 0.200 kg x 9.8 Nkg-1 x 0.30 m
• change in g.p.e. 0.59 J

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Complete
Answers
mass g ?h ?EP
3 kg 10 Nkg-1 400 cm 120 J
200 g 1.6 Nkg-1 30 m 9.6 J
7 kg 10 Nkg-1 4000 m 280 kJ
2000 g 24 Nkg-1 3000 mm 144 J
3 kg
1.6 Nkg-1
4000 m
144 J
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Falling objects

• If there is no significant air resistance then
  the initial GPE of an object is transferred into
  kinetic energy.
• ?EK ?EP
• ½ m v2 m g ?h

gpe mg?h
ke 0
?h
gpe ke
gpe ½ mg?h
ke ½ mv12
gpe 0
ke ½ mv22
ke mg?h
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Question

• A child of mass 40 kg climbs up a wall of height
  2.0 m and then steps off. Assuming no significant
  air resistance calculate the maximum
• (a) gpe of the child
• (b) speed of the child
• g 9.8 Nkg-1

• (a) max gpe occurs when the child is on the wall
• gpe mg?h
• 40 x 9.8 x 2.0
• max gpe 784 J
• (b) max speed occurs when the child reaches the
  ground
• ½ m v2 m g ?h
• ½ m v2 784 J
• v2 (2 x 784) / 40
• v2 39.2
• v ?39.2
• max speed 6.3 ms-1

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Power (P)

• Power is the rate of transfer of energy.
• power energy transfer
• time
• P ?E
• ?t
• unit watt (W)
• power is a scalar quantity

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• Power is also the rate of doing work.
• power work done
• time
• P ?W
• ?t

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Question 1

• Calculate the power of an electric motor that
  lifts a mass of 50 kg upwards by 3.0 m in 20
  seconds.
• g 9.8 Nkg-1

• ?EP m g ?h
• 50 kg x 9.8 Nkg-1 x 3 m
• 1470 J
• P ?E / ?t
• 1470 J / 20 s
• power 74 W

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Question 2

• Calculate the power of a car engine that exerts a
  force of 40 kN over a distance of 20 m for 10
  seconds.
• W F s
• 40 kN x 20 m
• 40 000 x 20 m
• 800 000 J
• P ?W / ?t
• 800 000 J / 10 s
• power 80 000 W

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Complete
Answers
energy transfer work done time power
600 J 600 J 2 mins 5 W
440 J 440 J 20 s 22 W
28 800 J 28 800 J 2 hours 4 W
2.5 mJ 2.5 mJ 50 µs 50 W
600 J
5 W
440 J
20 s
28 800 J
28 800 J
2.5 mJ
50 W
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Power and velocity

• power work done / time
• but work force x displacement
• therefore power force x displacement
• time
• but displacement / time velocity
• therefore
• power force x velocity
• P F v

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Question

• Calculate the power of a car that maintains a
  constant speed of 30 ms-1 against air resistance
  forces of 2 kN

• As the car is travelling at a constant speed the
  cars engine must be exerting a force equal to
  the opposing air resistance forces.
• P F v
• 2 kN x 30 ms-1
• 2 000 N x 30 ms-1
• power 60 kW

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Energy efficiency

• Energy efficiency is a measure of how usefully
  energy is used by a device.

useful energy transferred by the device
efficiency
total energy supplied to the device
As the useful energy can never be greater than
the energy supplied the maximum efficiency
possible is 1.0
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• Also

In all cases
percentage efficiency efficiency x 100
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Complete
Answers
Input energy (J) Useful energy (J) Wasted energy (J) Efficiency Percentage efficiency
100 40
250 50
50 0.20
80 30
60 60
60
0.40
40
200
0.80
80
10
40
20
24
56
0.30
120
0.50
50
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Internet Links

• Reaction time stopping a car - also plots
  velocity/time graph - NTNU
• Car Accident Reaction Time - NTNU
• Work (GCSE) - Powerpoint presentation by KT
• Kinetic Energy (GCSE) - Powerpoint presentation
  by KT
• Gravitational Potential Energy (GCSE) -
  Powerpoint presentation by KT
• Energy Skate Park - Colorado - Learn about
  conservation of energy with a skater dude! Build
  tracks, ramps and jumps for the skater and view
  the kinetic energy, potential energy and friction
  as he moves. You can also take the skater to
  different planets or even space!
• Rollercoaster Demo - Funderstanding
• Energy conservation with falling particles - NTNU
  
• Ball rolling up a slope- NTNU

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Core Notes from Breithaupt pages 148 to 159

1. What is the principle of conservation of energy?
2. Define work and give its unit. Explain how work
   is calculated when force and distance are not in
   the same direction.
3. With the aid of a diagram explain how work can be
   found from a graph.
4. Explain what is meant by, and give equations for
   (a) kinetic energy (b) gravitational potential
   energy.

• In terms of energy explain what happens as a body
  falls under gravity.
• In terms of energy and work define power.
• Show that the power of an engine is given by P
  Fv.

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Notes from Breithaupt pages 148 to 150Work and
energy

• What is the principle of conservation of energy?
• Define work and give its unit. Explain how work
  is calculated when force and distance are not in
  the same direction.
• With the aid of a diagram explain how work can be
  found from a graph.
• Try the summary questions on page 150

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Notes from Breithaupt pages 151 152Kinetic and
potential energy

1. Explain what is meant by, and give equations for
   (a) kinetic energy (b) gravitational potential
   energy.
2. In terms of energy explain what happens as a body
   falls under gravity.
3. Repeat the worked example on page 152 this time
   where the track drops vertically 70 m and the
   train has a mass of 3000 kg.
4. Try the summary questions on page 152

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Notes from Breithaupt pages 153 154Power

• In terms of energy and work define power.
• Show that the power of an engine is given by P
  Fv.
• Repeat the worked example on page 154 this time
  where the engine exerts a force of 50 kN with a
  constant velocity of 100 ms-1.
• Try the summary questions on page 154

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Notes from Breithaupt pages 155 156Energy and
efficiency

• Try the summary questions on page 156

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Notes from Breithaupt pages 157 to 159Renewable
energy

• Try the summary questions on page 159