Download Packet 9: Mechanical Energy

Unit III: Energy
Subunit A: Mechanical Energy
Equation
When to use it
Unit III: Energy
Subunit A: Mechanical Energy
Objectives
By the time we finish the labs and related materials in this unit, you should be able to:
1. Use Hooke’s Law to determine the spring force and analyze elastic energy systems.
2. Recognize and identify energy storage mechanisms: gravitational, kinetic, elastic,
dissipated.
3. Recognize the universal, fundamental nature of energy as opposed to different forms of
energy.
4. Make the distinction between energy storage and energy transfer.
5. Determine the quantity of kinetic energy, elastic potential energy, gravitational potential
energy, and frictional dissipated energy during an interaction.
6. Recognize and identify modes of energy transfer: working and heating.
7. Explain work as:
a. Energy transfer to/from a system via an external force.
b. Occurring when a force moves an object through a distance (force must be parallel to
the motion).
c. The area under an F vs. x graph.
8. Designate a system and use bar graphs to analyze a system in terms of energy storage and
transfer.
9. Define power as rate of energy transfer or usage; calculate power in Watts.
Unit III-A: Mechanical Energy
Worksheet 1
1. The graph at right was made from data collected during an
investigation of the relationship between the amount two
different springs stretched, when different forces were applied.
A) For each spring determine the spring constant.
B) For each spring, compare the amount of force required to
stretch the spring 3.0 m.
C) For each spring, compare the amount of force required to stretch the spring 10.0 m.
2. You are playing with you slinky on your stairs and decide to use it to weigh your physics
book. If the slinky has a spring constant of 20 N/m and your slinky stretches 0.5 m, what is the
weight of your book?
3. A 70-kg stuntman is attached to a bungee cord with an un-stretched length of 15 m. He
jumps off a bridge and comes to a stop just above the river, giving the bungee cord a stretched
length of 44 m. What is the spring constant of the bungee cord?
4. A spring with a force constant of 600 N/m is used on a scale for weighing fish. What is the
weight of a fish that would stretch the spring by 7.5 cm from its normal length?
5. The spring below has a spring constant of 10. N/m. If the block is pulled 0.30 m horizontally to
the right, and held motionless, what force does the spring exert on the block? Sketch a force
diagram for the mass as you hold it still. (Assume a frictionless surface.)
6. A spring in a pogo stick is compressed 12 cm when a 40 kg girl stands on the stick. What is
the spring constant for the pogo stick spring?
7. An orthodontist stretches a rubber band with a spring constant of 16 N/m around the braces
of a poor, unfortunate sophomore. If the rubber and stretches by 1.5 cm, what force does it
exert on the tooth?
8. An archer pulls her bow string back 0.400 m by exerting a force that increases uniformly from
zero to 230 N. What is the spring constant of the bow?
Unit III-A Reading: Energy
Energy- A conserved, substance-like quantity with the capability to produce change.
The idea of energy is an invention that proves very useful. Energy can be moved around and
stored in a variety of ways, but the energy itself is unchanged. Energy is universal and it does
not come in different "kinds" or exist in different "forms." There are many mechanisms for
energy storage. When something elastic is stretched, it has the potential do produce a change
which we call Elastic Energy, Ee. When an object is lifted above the ground, it has a potential
to fall which we call Gravitational Potential Energy, Eg. Energy is also stored in the chemical
bonds between atoms and molecules, which we call Chemical Potential Energy, Echem. Also,
energy can be stored in the motion of an object, in which case it is called Kinetic Energy, Ek,
because a moving object has the ability to produce a change (like when it crashes into
something!).
In all these cases, the energy can be easily retrieved. However, when energy is stored in the
random motion of molecules or the wave motion of molecules, it is very difficult to retrieve.
Friction within a system often causes energy to be transferred in the form of heat and sound
waves. In this unit, we will discuss the conservation of Mechanical Energy, including elastic
energy, gravitational energy, and kinetic energy, and lump together heat and wave energy as
Dissipated Energy, Ediss. We will later discuss heat and wave energy, but will not cover
chemical energy in this course.
A numerical amount of energy can be calculated for each storage mechanism. As energy is
transferred from one method of storage to another, the total amount of energy stays constant
(energy is conserved). When examining energy transfers, it is helpful to choose what methods
of energy storage are in our system and what methods are outside our system. Generally, the
smallest system that contains all the needed ways of storing energy is the easiest. Transferring
energy from one storage method to another or transferring energy into or out of a system is the
process of doing “Work.” The amount of work done is equal to the change in the energy storage
mechanisms.
Work in the physics sense has a very different meaning than in the traditional sense. Because
the physics definition of working is transferring energy, work is only done when something
change is produced or something is given a greater potential to produce a change. Even though
studying and thinking or even pushing with all your might against a wall may seem like work,
there is no transfer of energy taking place so no work is done. Mechanical work is only done
when a force exerted causes a change in position of an object, as energy storage usually
depends on position (e.g. height above the ground, displacement of a spring from equilibrium
position, or how fast it is moving).
Money analogy: We will define "the system" as the personal and institutional places where you
keep your money. You can store your money in a number of ways, in a checking account,
savings account, cash in a piggy bank, or a stock mutual fund. As you transfer money from cash
and savings to checking, the amount of money stays the same (is conserved) even though the
money is now stored somewhere else. Some transfers cost you money, such as using a credit
card or getting a cash advance. In this case the money is transferred out of your account and
into the bank's account. The money still exists, you just can't have it anymore. (This is like
dissipated energy.) Whenever you transfer money, you “put it to work” for you by exercising it’s
buying power. In these transfers, the money is not destroyed, but has simply changed hands.
Energy is a scalar (amount only, no direction) so it can't be represented with the arrows we used
for vectors. Instead, we will use bar graphs that show how much energy is stored in each
mechanism. The initial stored energy of a system is represented with a bar for each storage
mechanism and whether or not energy is transferred in or out of the system by working. Another
bar graph indicates the final distribution of energy stored within the system. To use the
representation for quantitative analysis, an equation or energy value can be written below each
bar. The sum of the initial stored energy in the system plus or minus any energy flowing in or out
of the system equals the sum of the final stored energy in the system.
Label the energy storage mechanisms in the following situations.
Situation
Energy Storage Mechanism
A dart is loaded into a dart gun.
A car is idling at a stoplight.
A bowling ball rolls down the lane toward the pins.
A camper warms his hands by the fire.
A weightlifter holds a barbell over his head.
A mouse trap is set with some cheese.
You are losing your hearing at a rock concert.
A rock climber hangs from a cliff face.
A baseball is thrown into the air.
Is work being done in the following situations? If so, what type(s) of energy is(are) changing?
State what types are increasing or decreasing.
Situation
A teacher applies a force to a wall and becomes exhausted.
A book falls off a table and free falls to the ground.
A waiter carries a tray full of meals above his head by one arm straight
across the room at constant speed.
A rocket accelerates through space.
A car slams on its brakes.
Energy
Mechanisms
Unit III-A: Mechanical Energy
Worksheet 2
Use bar graphs to analyze the energy changes in each situation given.
 Designate your choice of system,
 The graphs should be accurately divided among the energy storage mechanisms involved.
 Remember the 3 energy questions in deciding about energy changes:
1. Where does the energy come from?
2. What does the energy do?
3. Where does the energy go?
1. A ball is held above the ground, and then is dropped so it falls straight down.
(Restrict your analysis to the ball moving in the air, BEFORE it hits the ground.)
2. A wind-up toy is wound up, then "walks" across a table and comes to a stop.
3. A baseball is thrown up in the air and then falls back down. Place velocity vectors beside
each image of the baseball in the drawing, and do a graph for each position.
4. A bouncy ball is dropped and bounces up and down. Do a graph for each position of the ball
shown. Why does the ball not bounce as high each time? Where did the energy "go"?
5. An object rests on a coiled spring, and is then launched upwards.
6. A piece of clay is dropped to the floor.
7. 4. A ball rolls to a stop on the
floor.
Unit III-A: Mechanical Energy
Worksheet 4
Suppose in the lab one group found that F = 1000 N/m (Δx). Construct a graphical
representation of force vs. displacement. (Hint: Make the maximum displacement 0.25 m)
1. Graphically determine the amount of energy
stored while stretching the spring described above
from
A) x = 0 to x = 0.10 m.
B) x = 0.15 to x = 0.25 m
C) Determine the amount that the spring needs to be stretched in order to store 50 Joules of
energy.
2. A 25-kg box is held 1.5 m above the floor. A) What type of energy does it have? B) Calculate
how much energy it has.
3. How much work is done on a cart that is pushed 4 m across the floor by a horizontal 50 N
net force?
4. A 500-kg motorcycle traveling at a speed of 17.5 m/s. A) What type of energy does it store?
B) Calculate how much energy it has. C) How much energy will it have if it travels at twice the
speed?
5. Calculate the change in energy of 8,000,000 kg of water dropping 50 m over Niagara Falls.
6. Two students do 30,000 J of work to push a car a distance of 50 meters. How much force did
they exert on the car?
7. A boat (m=278kg) on the water has 167,664J of kinetic energy. How fast is it moving?
8. A spring stores 9783 J of potential energy when it is compressed by 0.34 m. What is the
spring constant value for this spring?
10. A person pulls a 50 kg box as shown. How much is the person applying the force working by
pulling the box a distance of 10.0 m?
Unit III-A: Mechanical Energy
Worksheet 3
For each situation shown below:
1. Describe your choice of system, unless it is specified for you (Is there friction? Is there a
spring involved? etc.)
2. Sketch an energy bar graph for the initial situation.
3. Then complete the analysis by completing the final energy bar graph.
4. A person pushes a car, with the parking brake on, up a hill.
5. A load of bricks rests on a tightly coiled spring, then is launched into the air.
6. A crate is propelled up a hill by a tightly coiled spring.
7. A bungee jumper falls off the platform and reaches the limit of the stretch of the cord.
8. An elevator, initially moving downward, is brought to rest on the ground floor.
9. A car, having just run out of gas, coasts up the hill to a stop.
Unit III-A: Mechanical Energy
Worksheet 5
1. A 60 kg student dives from a 10-meter platform into the pool below.
A) Determine her Eg at the top of the platform and label it
on the initial bar graph.
B) How much Ek does she possess at impact? Label this
on the final bar graph. Now calculate her velocity at impact.
2. How much work must be done by the brakes of a 2000 kg SUV to bring it to a stop if it was
traveling 11 m/s (25 mph)?
3. How much kinetic energy will a 3 kg physics book attain just before hitting the ground if it is
dropped from 1.5 meters above the ground?
4. A bungee cord has a spring constant of 140 N/m. How much work was done to stretch the cord
10 meters?
5. How much elastic energy does a bow need to store to shoot a 0.050-kg arrow at 25 m/s?
6. How much kinetic energy is required to launch a 2 kg watermelon 5 meters into the air?
Unit III-A: Mechanical Energy
Worksheet 6
1. A cart moving at 5 m/s collides with a spring. At the instant the cart is motionless, what is the
largest amount that the spring could be compressed? Assume no friction.
A) Define the system, then complete the energy bar graphs qualitatively.
B) Quantitative Energy Conservation Equation:
C) Determine the maximum compression of the spring.
2. A cart starts from rest at the top of a hill, and is bumped down the rough hill.
A) Calculate the initial amount of energy stored in the cart.
B) Determine final velocity of the cart at the bottom, if 10% of the energy is dissipated by friction.
3. A rock is shot straight up into the air with a slingshot that had been stretched 0.30 m. Assume
no air resistance.
A) Quantitative Energy Conservation Equation:
B) Determine the greatest height the rock could reach.
4. The bullet strikes a block of wood, which exerts, on average, a force of 50,000 N opposing
the motion of the bullet.
A) Quantitative Energy Conservation Equation:
B) Determine how far the bullet penetrates.
5. A baseball (m = 0.140 kg) traveling at 30 m/s moves a fielder’s glove backward 0.35 m when
the ball is caught.
A) Quantitative Energy Conservation Equation:
B) Determine the average force exerted by the ball on
the glove.
6. A spring whose spring constant is 850 N/m is compressed 0.40 m. What is the maximum
speed it can give to a 0.50 kg ball?
A) Quantitative Energy Conservation Equation:
B) Determine the maximum speed.
8. A bullet with a mass of 0.010 kg is fired from a rifle with a barrel that is 0.85 m long.
Assuming that the force exerted by the expanding gas to be a constant 5500 N, what speed
would the bullet reach?
9. A 24 kg child descends a 5.0 m high slide and reaches the ground with a speed of 2.8 m/s.
How much energy was dissipated due to friction in the process?
10. Remember the Wile E. Coyote shot from cannon problems? Suppose a robust 40 kg Wile E.
was shot straight up with an initial velocity of 50 m/s.
A) Assuming no energy is dissipated, what is the maximum height he could reach?
B) Suppose that 20% of his initial Ek were lost due to friction with the air (air resistance). Now
what is the maximum height he could reach?
Unit III-A: Mechanical Energy
Worksheet 7
1. A 60-kg box is lifted by a rope a distance of 10 meters straight up at constant speed. How
much power is required to complete this task in 5 seconds?
2. Hulky and Bulky are two workers being considered for a job at the UPS loading dock. Hulky
boasts that he can lift a 100 kg box 2.0 meters vertically, in 3.0 seconds. Bulky counters with his
claim of lifting a 200 kg box 5.0 meters vertically, in 20 seconds. Which worker has a greater
power rating?
3. A 2000 Ford Mustang GT is driving down a road with a constant speed of 30 m/s. The engine
must exert a 6465 N force to maintain this speed against air resistance and friction.
A) What is the power rating of the engine?
B) How does the Mustang’s power rating compare to the 242 hp Dodge Stealth engine
(1 hp = 746 W)?
4. An 82 kg boy scout hikes a trail up a mountain. During a 70-minute period, the hiker’s vertical
elevation increases by 623 meters.
A) Calculate the climber’s change in Eg.
B) Find the power generated to increase the hiker’s Eg.
5. How long would it take a 7.5 KW motor to raise a 500 kg piano to an apartment window 10
meters above the ground?
6. The trains on the Viper are raised from 10 m above ground at the loading platform to a height
of 60 m at the top of the first hill in 45 s. Assume that the train (including passengers) has a
mass of 2500 kg. Ignoring frictional losses, what power motor would be required to accomplish
this task?
7. A fire truck requires a pump that can move 150 kg of water 20 meters vertically each second.
Does a 5 HP pump have enough power to do the job? Why or why not? If not, what is the HP
rating of a pump that would do the job?
8. Your electric utility company sends you a monthly bill informing you of the number of kilowatthours you have used that month.
A) Is the utility charging you for energy or power? Explain.
B) How many joules does your 1600 W blow drier transfer if you dry your hair in 5.0 min?
Unit III-A: Mechanical Energy
Review Worksheet
Equations
k = Spring Constant, x = Displacement
k = Spring Constant, x = Displacement
F = Force, x = Displacement
m = Mass, v = Velocity
m = Mass, g = 10 m/s2, h = Height
F= Force or Friction, x = Displacement
W = Work, t = time, F = Force, v = Velocity
Spring Force
Elastic Energy
Work
Kinetic Energy
Gravitational
Dissipated
Power
Fs= kx
Ee= ½ kx2
W = Fx
Ek = ½ mv2
Eg = mgh
Ediss= Fx
P = W/ t or Fv
Matching.
A) Energy
B) Joule
C) Kinetic Energy
D) Law of Conservation of Energy
E) Potential Energy
F) Power
N
J, Nm
J, Nm
J, Nm
J, Nm
J, Nm
W, J/s
G) Watt
H) Work
1.___________ The SI unit of work and energy is the _____.
2.___________ Energy that is stored in an object by virtue of its position is _____.
3.___________ Within a closed, isolated system, energy can change form, but the total amount
of energy is constant.
4.___________ The form of energy found in moving objects is _____.
5.___________ The ability of an object to produce a change in itself or its surroundings is _____.
6.___________ The rate of doing work is _____.
7.___________ The transfer of energy by mechanical means is _____.
8.___________ The unit of power equal to one Joule per second is the _____.
Multiple Choice
9. When a force is exerted on an object, work is done only if the object
A) is heavy
B) remains stationary
C) moves
D) burns
10. In which of the following situations is NO work done on a football?
A) Picking it up
B) Carrying it down the field C) Dropping it
D) All of the above
11. How much work is done if you raise a 6.0 N weight 1.5 m above the ground?
A) 0 J
B) 4 J
C) 7.5 J
D) 9 J
12. Which of these is not needed to find the kinetic energy of an object?
A) Shape
B) Mass
C) Velocity
D) None of these
13. When you throw a ball up into the air, the total energy of the ball at any point in its flight can
be expressed by the kinetic energy _______ the potential energy.
A) times
B) minus
C) divided by
D) plus
14. As an object falls towards Earth, the potential energy of the object
A) increases
B) decreases
C) remains the same.
15. The total amount of energy in an isolated, closed system
A) is increasing
B) is decreasing
C) remains constant D) cannot be measured
16. The gravitational potential energy of an object close to Earth is directly related to
A) its mass
B) its height above Earth
C) its velocity
D) both a and b
17. The transfer of energy by mechanical means is
A) force
B) work
C) effort
D) acceleration
Practice Problems
The following data were collected for two springs:
18. What are the spring constants of springs 1 and 2?
19. How much Eel would be stored if spring 2 were stretched from 0 to 0.40 m?
21. A bowling ball pendulum (m = 5 kg) is pulled up to a height of 0.5 m above
the ground. It is then released, and starts to move. At the lowest point of its
swing, it is nearly touching the ground (Intermediate situation). After swinging
down, it begins to go back up, then comes to a stop at its highest point (Final
situation), before swinging back down.
A) Describe your choice of system (and state whether or not there is friction).
B) Draw bar graphs for the initial,
intermediate and final situations.
C) Find the speed of the bowling ball
at the bottom of its swing.
22. A 1000 kg car is traveling at a constant speed of 30 m/s.
A) How much energy is dissipated as the car comes to rest?
B) If the car stops in 100 meters, what is the average force applied to the car?
23. A 6 kg cat is napping on the top of a refrigerator when he rolls over and falls. He has 86 J of
kinetic energy just before he lands on his feet on the floor. How tall is the refrigerator?
24. Hans and Franz are moving 800 N cartons from the floor up onto a 1-m platform. Hans
moves 10 cartons in 7 minutes. Franz moves the same number of cartons in 5 minutes. Which
worker does more work? Which has more power?