Download Conservation of Energy, Power and Efficiency

SPH3U
Lesson 02
Energy
CONSERVATION OF ENERGY
LEARNING GOALS
Students will:
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Investigate and apply the law of conservation of energy.
INVESTIGATING CONSERVATION OF ENERGY
START THE PhET SIMULATION “ENERGY SKATE PARK: BASICS”:
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go to http://phet.colorado.edu
Click on “Play with Sims”. Choose “Physics”, then choose “Work, Energy & Power”. Run
the simulation “Energy Skate Park: Basics”.
Click on the tab “Track Playground”. Make a simple U-shaped track with equal heights on
both sides.
On the right hand side, click on the “Bar Graph” option.
Place the skater on the ramp and let him go.
QUESTIONS
1.
2.
Watch the skater for a while.
a)
How does his maximum height on the right side of the ramp compare to his
maximum height on the left side of the ramp?
b)
How does his maximum height after several back and forth oscillations compare to
his maximum height at the start? Is this realistic? What is missing from this
simulation?
c)
Click on the FRICTION tab on the top. On the right hand side, turn on friction and
answer the questions above again.
TURN FRICTION OFF. Start the skater near the top of one side of the ramp. Watch the
bar graph on the right.
a)
Where is the kinetic energy the highest? Lowest?
b)
Where is the potential energy the highest? Lowest?
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Lesson 02
c)
3.
Energy
Look at the Total Energy bar. What is happening to it? Click on the pause button.
With a ruler, measure the height of all three energy bars (kinetic, potential and
total). How do they compare?
TURN FRICTION ON and using the slider, set it to a low value. Start the skater near the
top of one side of the ramp.
a)
What new energy now appears in the bar graph? What is causing this energy to
appear?
b)
Click on pause. Measure all 4 energy bars with a ruler. What conclusion can you
make?
c)
Un-pause the simulation and let it run until the skater is stopped. Describe what
has happened to the energy.
4.
Throughout ALL the simulations above, what did you notice about the TOTAL ENERGY bar?
Explain why you think this occurs.
5.
a)
Your answer to 4. above is called the Law of Conservation of Energy. State the
LAW OF CONSERVATION OF ENERGY (Google it to see what it says).
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Lesson 02
Energy
b)
In the simulations in which friction was off, describe what was happening to the
energy.
c)
In the simulations where friction was turned on, describe what was happening to
the energy.
Modify your track so that it is still U-shaped, but one side is higher than the other. Turn
friction off.
a)
Start the skater at the highest point of the LOW SIDE of your track. Describe his
maximum height on the high side. Why does he reach this height?
b)
Start the skater at the highest point on the HIGH SIDE. Release him/her.
Describe what happens when he/she gets to the other side and why.
c)
Turn friction on. Repeat a) and b) above. Describe your results.
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Lesson 02
Energy
CONSERVATION OF ENERGY
The law of conservation of energy states that energy cannot be created or destroyed. If there is
no friction, then this means that the total mechanical energy of an object must be the same at
all times.
ETOTi = ETOTf
Eki + Egi = Ekf + Egf
EXAMPLES
EXAMPLE
Fill in the missing energies. KE = kinetic energy PE = potential energy
EXAMPLE
Consider the roller coaster shown.
1. Assume it is at rest at point A.
Write an equation for the total
mechanical energy of the coaster
at this point.
2. Point D is 70 m above the ground. Write an equation for the total energy of the ride at this
point.
3. Since total mechanical energy must be conserved, write one big equation that relates the
total energy at A to the total energy at B
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Lesson 02
Energy
4. Use your equation in 3 to determine the speed of the coaster at the top of the loop (point D)
(31 m/s)
5. Use a similar process to determine the speed of the roller coaster at point E. Explain the
difference in speed between points D and E using energy concepts. (40. m/s)
EXAMPLE
A 665 kg roller coaster car starts from rest at the start of the first drop. It rolls down to the
bottom of the first drop which is vertically 25 m lower than the top. The actual length of the
track from the top to the bottom is 35 m (an angle of 450 from the vertical).
a)
Assuming that friction is negligible, determine the speed of the roller coaster when
it reaches the bottom of the first drop.
b)
If the force of friction is 691 N, what is the speed of the roller coaster car at the
bottom of the first drop?
SOLUTION
a)
Use the lowest point (the bottom of the drop) as your reference point. The total
energy of the roller coaster car will be gravitational potential only as the car is
initially at rest.
ET = Eg = mgh = (665 kg)(9.8 m/s2)(25 m) = 1.63 x 105 J
At the bottom of the drop, you are at the reference point, so gravitational potential
energy is 0. All the energy is now kinetic.
ET = Ek = ½ mv2
1.63 x 105 J = ½ (665 kg) v2
v = 22 m/s
b)
Here, the total energy becomes kinetic and heat due to friction.
ET = Ek + Wf = ½ mv2 +Ff d
Therefore
1.63 x 105 J = ½ (665 kg) v2 + (691 N)(35 m)
v = 20. m/s
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Lesson 02
Energy
PRACTICE PROBLEMS
1. A 4.0 x 104 kg roller coaster starts from
rest at point A. Neglecting friction,
calculate its potential energy relative to
the ground, its kinetic energy and its
speed at points B, C and D. (278 m/s,
12 m/s, 19 m/s)
2. A 2.5 kg lead ball and a 55 g piece of lead shot are both dropped from a height of 25. m.
Neglecting air friction, what is the kinetic energy of each object just before it hits the ground?
What is the velocity of each object just before it hits the ground? (22 m/s, 22m/s)
3. You throw a ball directly upward, giving it an initial velocity of 10.0 m/s. Neglecting friction,
what would be the maximum height of the ball? (5.1 m)
4. A 32 kg crate was pushed down a frictionless ramp. Its initial velocity at the top of the ramp
was 3.2 m/s. Its velocity when it reached the bottom of the ramp was 9.7 m/s. The ramp
makes an angle of 25° with the horizontal. How long is the ramp? (10. m)
5. A wrecking ball, with a mass of 315 kg, hangs from a crane on 10.0 m of cable. If the crane
swings the wrecking ball so that the angle that the cable makes with the vertical is 30.0°,
what is the potential energy of the wrecking ball in relation to its lowest position? What will
be the kinetic energy of the wrecking ball when it falls back to the vertical position? What
will be the speed of the wrecking ball? (5.1 m/s)
6. A 2.00 kg mass is attached to a 3.00 m string and is raised at an angle of 45°
relative to the rest position as shown. Calculate the gravitational potential
energy of the pendulum relative to its rest position. If the mass is released,
determine its velocity when it reaches its rest (vertical) position. (4.2 m/s)
7. A worker on the roof of a building that is under construction dropped a 0.125
kg wrench over the edge. Another workman on the eighth floor saw the wrench go by and
determined that its speed at that level was 33.1 m/s. The first floor of the building is 12.0 m
high and each successive floor is 8.00 m high. Neglecting air friction, how many floors does
the building have? How fast was the wrench falling just before it hit the ground? What was
its kinetic energy just before it hit the ground? (14 floors, 48 m/s, 140 J)
8. A 0.50 kg baseball falls from a 2.3 m shelf onto the floor, than bounces up to a height of 1.4
m before you catch it.
a. Calculate the gravitational potential energy of the ball before it falls. (11 J)
b. Ignoring frictional effects, determine the speed of the ball as it strikes the floor, assuming
that it fell from rest. (6.7 m/s)
c. How fast is the ball moving just as before you catch it? (4.2 m/s)
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Lesson 02
Energy
POWER
LEARNING GOALS
Students will learn:
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Power is the rate at which work is done (rate of energy transfer).
POWER
Power is essentially how fast energy is transferred or changed.
UNIT
A.
What is the unit of power using the units of energy and time?
B.
What is the unit of power using only m, kg and s?
C.
What is official name of the unit of power and its metric symbol?
ALTERNATE POWER CALCULATION
Using the formula for power:
Substitute in the formula for Work. You will notice that power can be expressed as a force
multiplied by another quantity (look at the remaining terms to determine that quantity).
A.
Write the alternate formula for power:
EXAMPLES
EXAMPLE 1
If a car engine does 25,000 kJ of work over 12 s to climb a hill, calculate the average power of
the engine. (2.1 x 106 W)
EXAMPLE 2
A 90 kW motor exerts an average force of 4.0 kN to move a speedboat at a constant speed
through the water. How fast, in km/h, is the boat moving? (81 km/h)
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Lesson 02
Energy
EXAMPLE 3
A snow tubing lift pulls riders up a 120 m long hill at a constant speed. The hill makes an angle
of 22o to the horizontal and there is 145 N of friction between the tubes and the snow. The rider
is 55 kg and it takes 30.0 s to get up the hill.
a) Sketch a labeled diagram:
b) How much work does friction do?
c) What will be the gain in the rider’s gravitational potential energy?
d) How much work will the lift need to do?
e) What must be the power output of the motor in order for the rider to get up the hill in
30.0 s?
a) 17 400 J b) 24 254 J c) 41654 J
d) 1388 W
PRACTICE PROBLEMS
1. A fully outfitted mountain climber, complete with camping equipment, has a mass of 85
kg. If the climber climbs from an elevation of 2900 m to 3640 m in exactly one hour,
what is the climber’s average power? (170 W)
2. An elevator motor provides 32 kW of power while it lifts the elevator 24 m at a constant
speed. If the elevator’s mass is 2200 kg, including the passengers, how long does the
motion take? (16 s)
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Lesson 02
Energy
3. Molly, a Clydesdale mare, exerts a force of 670 N to pull a farm tractor at a speed of 4.00
km/h along a road for 2.00 h. How much work does Molly do? What is Molly’s power?
(5.4 MJ, 740 W)
4. Lance Armstrong is accelerating up one of the hills on his bike in The Tour de France. The
combined mass is 80.0 kg. The hill is 650 m long and the total elevation change of the
hill is 25 m. His speed at the bottom of the hill is 2.0 m/s. Lance reaches the top of the
hill in 150 seconds and is traveling at 7.5 m/s. If the average force of friction on the bike
wheels is 35 N, find the average power exerted by Lance. (300 W)
5. A 1200 kg Toyota Echo is accelerating down a hill. The hill is 150 m long and the total
decline in elevation of the hill is 15 m. The car accelerates from a speed of 2.0 m/s at the
top to an unknown velocity at the bottom in 11 s. The power of the car is 9.6 kW, while
the average retarding force of friction is 600 N. Find the speed of the car at the bottom of
the hill. (18 m/s)
EFFICIENCY
LEARNING GOALS
Students will learn:
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Machines are not perfectly efficient because of energy losses to heat.
EFFICIENCY
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The efficiency of a machine or device is the extent to which it transforms input energy or
work into the intended type of output energy or work.
Efficiency can be expressed as a decimal or a percent.
Many of our homes use incandescent light bulbs for the purpose of lighting. However, you may be
surprised to learn that only about 5% of the power used by the bulb is actually transformed into
light. The rest is transformed into wasted thermal energy (… maybe you're not surprised if you've
ever made the mistake of touching a bulb that has been on for a while—it is HOT!) Therefore, 95% of
the money we spend to operate that light bulb is wasted! Incandescent lights are basically electric
heaters that give off some light as a by-product.
Because only 5% of the power put into a light bulb is put out as useful light, we say the bulb has
a percent efficiency of 5%. Efficiency and percent efficiency of an energy transformation device can
be evaluated in terms of total power input and useful power output. The method of measuring
these energies is simply to measure the electric current and voltage being supplied to the light (total
input power) and the light being emitted in lumens (useful output power).
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SPH3U
Lesson 02
Energy
SOME EXAMPLES OF KNOWN EFFICIENCIES
a) How does 100% get
turned into 5.7% in the
top diagram?
b) Is an electric car
engine more or less
efficient than a
combustion engine?
c) Assuming that the incandescent light and the fluorescent light in diagram (c) produce the
same amount of radiant energy, how does the diagram show that the fluorescent light is
more efficient?
d) What stages of the energy path shown in diagram (d) do you think are most efficient?
Least efficient? List all the sources of energy loss that you can think of at each step.
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Lesson 02
Energy
SECOND LAW OF THERMODYNAMICS
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Thermal energy is always transferred from a warm object to a colder object.
o An ice cube will absorb heat from liquid water and will melt in the water; the water
will get colder as a result of the heat loss. An ice cube will not become colder in
water thus making the water warmer.
In a steam engine, as thermal energy goes from ______- to ______-temperature objects,
some of the heat can do mechanical work (like move pistons).
The second law asserts that during the heat transfer, some energy will leave the system
without doing work.
No machine is 100% efficient. This includes the scams you see on youtube
about magnetic devices moving without energy being added to keep them going.
EFFICIENCY PROBLEM
Ramps are simple machines. They enable you to raise a heavy object a certain height with a
lower force. By increasing the distance over which work is done, the force can be reduced.
1.
Set up a ramp. Get one of the friction blocks you used in the friction lab. Place a
textbook on the block. Pull the block/textbook up the ramp with a spring scale at a
constant speed.
2.
How far along the ramp did you pull the block/book?
How much force did it take?
How much work did you do?
How much gravitational potential energy did the block/book gain? (Measure the
quantities you need to determine this).
3.
What is the efficiency of the ramp (what percentage of your work actually went into
gravitational potential energy of the block/book)?
4.
Where did the rest of your work go?
5.
How much force would it have taken to raise the book/block straight up by hand (not
using the ramp)?
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Lesson 02
Energy
ENERGY TRANSFORMATIONS WITH EXTERNAL FORCES
LEARNING GOALS
Students will learn:
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how external forces affect the energy of a system
to solve energy transformation problems with external forces.
CONSERVATION OF ENERGY IN NON-ISOLATED SYSTEMS
So far we have only been discussing situations where there is no friction to remove energy from
an object that is moving. Also, there have been no external forces that have added energy to
the object. This situation describes an isolated system – one where the object does not
interact with its surroundings. (In our case, the “system” is the object that is moving)
Think of a roller coaster. What two parts of the surroundings have we thus far assumed that it
does not interact with? _________________________________________________
However, in reality, objects have surroundings and they interact. Friction, air resistance and
applied forces are examples of external forces because they come from outside the system.
FRICTION
1. We know that friction removes energy from an object. Say you have a fireman sliding down
a fire pole that is 5.0 m long and the force of friction is 350 N.
a) How much work does friction do on the firefighter as he slides down?
So let’s calculate how fast the fireman will be going at the bottom of the pole.
Assume he starts at rest and has a mass of 70.0 kg.
b) Label the diagram with all know information.
c) What type(s) of energy does the firefighter have at the top of the pole?
Calculate his total energy at the top of the pole.
d) Since friction removed energy, how much energy does he have left at the bottom of the
pole?
e) What type(s) of energy does he have at the bottom of the pole? Find his speed there.
f)
The law of conservation of energy says that energy cannot be created or destroyed. Is
this still true in the situation you just did?
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Lesson 02
Energy
2. Remember that work is a transfer of energy. It changes the total energy of the system:
W = ETOT
W = ETOTf - ETOTi
a. Rearrange the equation above to solve for ETOTf
With friction, the work done is negative, so we often write
ETOTi – Wf = ETOTf
FORCES THAT ADD TO THE ENERGY
3. In some situations external forces add energy to an object.
Imagine an 1800 kg car going up a frictionless hill. At the
bottom of the hill, it is going 45 km/h (12.5 m/s) and the hill
makes an angle of 25.0o to the horizontal. The car maintains a
steady speed as it goes up the hill which is 255 m long.
a. Label the diagram.
b. Determine the total mechanical energy of the car at the bottom of the hill.
c. Determine the total mechanical energy of the car at the top of the hill.
d. Explain why they are different.
e. How much work did the car’s engine have to do on the car in order for it to maintain
its speed as it drove?
f.
Determine the average force the engine exerted as the car went up the hill.
4. Imagine the same situation as in question 3, but say the average force of friction is 450 N.
The car still maintains the same speed as it goes up the hill.
a. Will the friction make the engine have to do more work or less work? Why?
b. Determine the amount of work done by friction. (Is it positive or negative?)
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Lesson 02
Energy
c. Remember that net work is equal to the change in total energy. Use this idea to
determine the amount of work the engine had to do on the car in this case. Show all
your steps carefully.
PRACTICE
1. The Millennium Force, the tallest roller coaster in North America, is 94.5 m high at its highest
point. What is the maximum possible speed of the roller coaster? The roller coaster’s actual
maximum speed is 41.1 m/s. How much energy was lost as thermal energy due to friction?
What percentage of its total mechanical energy is this? (43 m/s, 8.9%)
2. A 15 kg child slides from rest, down a playground slide that is 4.0 m long. The slide makes a
40° angle with the horizontal. The child’s speed at the bottom is 3.2 m/s. What was the
force of friction that the slide was exerting on the child? (75 N [up the slide])
3. In the winter sport of skiing, some beginners stop by gliding up a small hill. In one particular
case a skier, of mass 55 kg, traveling at 10 m/s at the base of the hill comes to rest by going
up a hill with a height 2.0 m. The frictional force offered by the snow is 300 N. Find the
distance it takes the skier to stop. (5.6 m)
4. A 1400 kg car is accelerating up a hill. The hill is 150 m long and the total rise of the hill is
6.0 m. The car accelerates from a speed of 7.0 m/s at the bottom to 15 m/s at the top in 12
s. If the average retarding force of friction is 700 N, find the average work applied by the
car. (3.1 x 105 J)
Answers to Class Work: 1a)1.8 x 103 J c)3.4 x 103 J d)1.7 x 103 J e)6.9 m/s 3b)1.41 x
105 J c)2.04 x 106 J e)1.90 x 106 J f)7.46 x 103 J 4b)1.15 x 105 J c)2.02 x 106 J
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