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2.1d Mechanics
Work, energy and powerBreithaupt 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 θ
P = ΔW / Δt
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.
ΔE p = mgΔh
E k = ½ mv2
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Work (W )
Work is done when a force moves its point ofapplication.
work = force x distance moved in thedirection 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 θ
The point of application of force, F moves
distance s cos θ when the object moves
through the distance s .
F
s
θ object
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Question 1
Calculate the work done when a force of5 kN moves through a distance of 30 cm
work = force x d istance= 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.
wo rk = 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 cosθ
= 800 N x 50 m x cos
30°
= 40 000 x cos 30°
= 40 000 x 0.8660work = 34 600 J
wind force = 800 N
distance moved
by yacht = 50 m
30°
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Complete:
Force Distance Angle between
F and sWork
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
Answers
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 thecurve is equal to
the work done.
F
s
force
distance
area = work done
F
s
force
distance
area = work
= ½ F s
area = work
found by
countingsquares on
the graph
F
s
force
distance
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Question
Calculate the work done by
the brakes of a car if theforce exerted by the brakesvaries over the car’s brakingdistance of 100 m as shownin the graph below.
Work = area under graph
= area A + area B
= (½ x 1k x 50)
+ (1k x 100)
= (25k) + (100k)
work = 125 kJ
2
force / kN
distance / m
1
50 100
area B
area A
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Energy (E )
Energy is needed to move objects, to changetheir 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 ofenergy 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 energyKinetic energy (KE)
Energy due to a body’s motion. Potential energy (PE)
Energy due to a body’s position
Thermal energy
Energy due to a body’stemperature.
Chemical energy
Energy associated with chemical
reactions.
Nuclear energy
Energy associated with nuclearreactions.
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 (E K )
Kinetic energy is the energy an object hasbecause of its motion and mass.
kin etic energy = ½ x mass x (speed) 2
E K = ½ m v 2
Note: v = speed NOT velocity.
The direction of motion has no relevance to kineticenergy.
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Question 1
Calculate the kinetic energy of a car of mass800 kg moving at 6 ms-1
E K = ½ m v 2
= 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 itskinetic energy is 15 000J
E K = ½ m v 2
15 000J = ½ x 1200kg x v 2
15 000 = 600 x v
2
15 000 ÷ 600 = v 2
25 = v 2
v = 25
speed = 5.0 ms-1
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Question 3Calculate the braking
distance a car of mass900 kg travelling at aninitial speed of 20 ms-1 ifits brakes exert a constantforce of 3 kN.
k.e. of car = ½ m v 2
= 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 thiskinetic energy.
W = F s
180 000 J = 3 kN x s
180 000 = 3000 x ss = 180 000 / 3000
braking distance = 60 m
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Complete:
Mass Speed K inetic 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
Answers
8 kg
12 ms-1
1.5 x 1011 J
3.2 J
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Gravitational Potential Energy (gpe )
Gravitational potential energy is theenergy an object has because of itsposition in a gravitational field.
change in g .p .e.
= mass x gravi tat ional f ield strength
x change in height
ΔE P = m g Δh
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Question
Calculate the change in g.p.e. when a massof 200 g is lifted upwards by 30 cm.
(g = 9.8 Nkg -1 )
ΔE P = m g Δh= 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:
mass g Δh
ΔE P
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
Answers
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.
ΔE K = ΔE P
½ m v 2 = m g Δh
Δh
m
½ Δh
v 1
v 2
gpe = mg Δh
ke = ½ mv 2 2
ke = 0
gpe = 0
gpe = ke
gpe = ½ mg Δh
ke = ½ mv 1 2
ke = mg Δh
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Question A child of mass 40 kgclimbs up a wall of height2.0 m and then steps off. Assuming no significantair resistance calculate themaximum:
(a) gpe of the child
(b) speed of the child
g = 9.8 Nkg -1
(a) max gpe occurs whenthe child is on the wall
gpe = mg Δh
= 40 x 9.8 x 2.0
max gpe = 784 J
(b) max speed occurs whenthe child reaches the ground
½ m v 2 = m g Δh½ m v 2 = 784 Jv 2 = (2 x 784) / 40
v 2 = 39.2v = 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
t imeP = ΔE
Δt
unit: watt (W)
power is a scalar quantity
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Power is also the rate of doing work.
power = work done
t ime
P = ΔW Δt
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Question 1
Calculate the power of an
electric motor that lifts amass of 50 kg upwards by
3.0 m in 20 seconds.
g = 9.8 Nkg -1
ΔE P = m g Δh
= 50 kg x 9.8 Nkg-1 x 3 m
= 1470 J
P = ΔE /
Δt
= 1470 J / 20 s
power = 74 W
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Question 2Calculate 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 = ΔW / Δt
= 800 000 J / 10 spower = 80 000 W
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Complete:
energy
transfer
wo rk done t ime 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
Answers
600 J 5 W
440 J 20 s28 800 J 28 800 J
2.5 mJ 50 W
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Power and velocity
power = work done / t ime
but: work = force x disp lacement
therefore: power = force x disp lacement
t ime
but: disp lacement / t ime = veloci ty
therefore:
power = force x veloc i tyP = F v
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Question
Calculate the power of a car
that maintains a constantspeed of 30 ms-1 against air
resistance forces of 2 kN
As the car is travelling at a
constant speed the car’sengine 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 howusefully energy is used by a device.
efficiency =
useful energy transferred by the device
total energy supplied to the device
As the useful energy can never be greater
than the energy supplied the maximumefficiency possible is 1.0
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Also:
efficiency =useful work output
energy supplied
useful power outputefficiency =
power input
In all cases:
percentage efficiency = efficiency x 100
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Complete
Input
energy (J)
Useful
energy (J)
Wasted
energy (J)
Efficiency Percentage
efficiency
100 40
250 50
50 0.20
80 30%
60 60
60
200
10 40
24 56
120
0.80
0.50
0.30
20%
0.40
80%
50%
40%
Answers
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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 ofconservation of energy?
2. Define work and give its unit.Explain how work iscalculated when force anddistance are not in the samedirection.
3. With the aid of a diagramexplain how work can befound from a graph.
4. Explain what is meant by,and give equations for (a)kinetic energy & (b)gravitational potential energy.
5. In terms of energy explainwhat happens as a body fallsunder gravity.
6. In terms of energy and workdefine power.
7. Show that the power of anengine is given by: P = Fv .
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Notes from Breithaupt pages 148 to 150
Work and energy
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 notin the same direction.
3. With the aid of a diagram explain how work can
be found from a graph.
4. Try the summary questions on page 150
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Notes from Breithaupt pages 151 & 152
Kinetic and potential energy
1. Explain what is meant by, and give equationsfor (a) kinetic energy & (b) gravitationalpotential energy.
2. In terms of energy explain what happens as abody falls under gravity.
3. Repeat the worked example on page 152 this
time where the track drops vertically 70 m andthe train has a mass of 3000 kg.
4. Try the summary questions on page 152
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Notes from Breithaupt pages 153 & 154
Power
1. In terms of energy and work define power.
2. Show that the power of an engine is given by:
P = Fv .
3. 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.
4. Try the summary questions on page 154
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Notes from Breithaupt pages 155 & 156
Energy and efficiency
1. Try the summary questions on page 156
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Notes from Breithaupt pages 157 to 159
Renewable energy
1. Try the summary questions on page 159