Introduction to Energy
Energy is not only a major topic in physics but it is a source of endless discussion in the world today. Wars have been fought over energy. The politics of the world revolve around energy. The wealth of nations depends on their supply of energy. Even your standard of living, the clothes you have and the type of car you drive are dependent upon energy. In middle school you probably learned that there are all kinds of energy. Your food has chemical energy that becomes available to your body. There is nuclear energy that powers some power plants. There is solar energy. The list goes on and on. To make it simple, which believe it or not is the goal of physics, the physicist takes all types of energy and puts them into only two groups. All types of energy become part of one of these two basic types. These two basic types are;
Kinetic Energy: This is the energy that any moving object has. It is called the "energy of motion". When an object is moving it has kinetic energy. When it is stopped it does not have kinetic energy. The symbol we use in an equation for kinetic energy is Ek, to make us think of Energy, kinetic. To find kinetic energy use the equation
E
k= (1/2) mv
2.
Because mass is in kilograms and speed is in m/s, the unit on Ek will be kg*m2/s2. Recall that a Newton
is a kg*m/s2. Because you could multiply a Newton by a meter and get the E
k unit, you could express
energy in units of N*m. However, because the unit is used so often, it is renamed and called a Joule. The abbreviation for a Joule is J. So kinetic energy is always measured in Joules.
Potential Energy: This is the potential to have kinetic energy. If any object has the possibility of moving, then it has some potential to release kinetic energy. For example, a bomb has potential energy, because if the bomb is detonated pieces of the bomb will move. A diver on a diving board also has potential energy because when he leaves the board he will be pulled downward by gravity and his speed will increase. Because
the speed is not zero the person will have kinetic energy, so they must have had some potential to have kinetic energy earlier. In an equation, the symbol that we use for potential energy is Ep, to make us think of Energy, potential. There are many ways to find the potential energy and it depends on the type of potential energy for the exact equation. However, in physics we usually think of potential energy due to gravity and the equation for computing that is
E
p= mad .
In this equation m is the mass, a is the acceleration due to gravity (9.8 m/s2 on earth) and d is the
distance above the starting point (the vertical distance or the height). Because m is in kg, a is in m/s2 and
d is in meters, the unit of potential energy is Joules.
Whenever a new quantity is introduced in physics, the standard metric units associated with that quantity are discussed. In the case of energy (and also work), the standard metric unit is the Joule (abbreviated "J"). One Joule is equivalent to one Newton of force causing a displacement of one meter. In other words,
To help you understand how to do the calculations, here are a few examples;
Example #1: A 5.0 kg ball is moving at 3.0 m/s. What is the ball's kinetic energy? Given: m=5.0 kg
v = 3.0 m/s
Ek = ?
Equation: Ek= (1/2) mv2
Substitution: Ek= (1/2) 5.0 kg (3.0m/s)2= 2.5 (9) kg*m2/s2
Answer: 22.5 J = 23J
Example #2: A 6.0 kg rock is held 3.0 meters off the ground, about to be dropped. What is the potential energy of the rock?
Given: m = 6.0 kg
d = 3.0 m
Ep = ?
© Fyzics Softext, Inc. page 103
The Joule is the unit of energy (and
work).
Equation: Ep= mad
Substitution: Ep= (6.0kg)(9.8 m/s2)(3.0m) Answer: = 176.4 J = 180J
Now, suppose that the rock in example #2 is dropped. As the object gets closer to the ground the potential energy will get less because the distance above the ground is getting smaller. In fact, when the object hits the ground the potential energy will be zero because d will be zero. What happened to all that potential energy? Well, it made the object move. As it fell the speed of the object was increasing. What was the speed just before the object hit the ground?
Any object 3.0 meters off the ground will take
t
=
√
2
d
/
a
to hit the ground. So the object will taket
=
√
2
(
3.0
m
)/
9.8
m
/
s
/
s
to hit√
0.612244898
= 0.78246 seconds. Since vf= at this means that vf= (9.8 m/s2) (.78246s)= 7.6681 m/s. Do you see why this is important? Well it may not be immediately obvious so I will tell you. Use this v (the 7.6681 m/s) to find the kinetic energy. Ek= (1/2)(6.0 kg)(7.6681m/s)2= 176.399 J or 180 J !!! SURE- does that surprise you? The kinetic energy just before the rock hits the ground is exactly equal to the potential energy that the rock had before it was dropped! But you see that is the definition of potential energy- just before the rock hits it has lost all of the potential energy and converted it into kinetic energy- the motion. So you can see that the following will always be true if there are no outside influences;
Ep = Ek or in words; The change in the potential energy will equal the change in the kinetic energy in a closed system
( no outside influences).
The sum of the kinetic and potential energy is called the total energy of a system. In symbols, ET= Ep + Ek. Because any changes in the potential and kinetic energies are equal to each other but opposite in sign, the total energy of a closed system will remain constant. The key term here is "closed" system. Any outside influences may change the total energy. For example, when a pendulum is swinging, if there were no friction then it would continue to swing forever. However, because there is friction, this represents an outside influence and energy is being removed from the system so the total energy decreases and the bob stops swinging. Of course, to start the bob swinging some person probably reached in and gave it a push. This represented energy being put into the system. Any time that energy is added to a system a physicist says that work has been done. Work is defined as the change in total energy of a system. If energy is added to as system, positive work is done. If work is taken from a system, negative work is done. In an equation the symbol for work is W. Because work is a change in energy, the unit for work is the same as the energy unit, Joules. The equation is;
W= E
So when you start a pendulum bob swinging you do positive work on the bob. As the bob loses energy through swinging, the bob does negative work on the surroundings (the air gets hot and the string supporting the bob gets warm.) Because you had to push on the bob to start it, you had to apply some force through some distance. So another way to view work is W= Fd. In this equation W
is work, F is force, and d is the distance. There is one other fact you should know about work. The force and the distance must be in the same direction. So work is only done by the part of the force that is in the direction that the object moves.
Example #3: How much work is done in moving a mass 4.0 meters if you push on it with a force of 8.0 N the entire time?
Given: d = 4.0 m
F = 8.0 N
W = ?
Equation: W=Fd.
Substitution: W= (8.0 N)(4.0 m)
Answer = 32 N*m = 32J
Because both the force and the distance must be in the same direction, sometimes work is not done when you expect it to be done! Consider a waiter who carries a tray full of meals across the room. The force
The actual equation for Work includes a term for the calculation of the work if the Force and the displacement are in different directions. That equation is;
The equation for work lists three variables - force, displacement, and the cosine of the angle between them. The angle theta in the equation is associated with the amount of force which causes a displacement. When a force is exerted on an object at an angle to the horizontal, only a part of the force contributes to (or causes) a horizontal displacement.
Let's consider the force of a chain pulling upwards and rightwards upon Fido in order to drag Fido to the right. It is only the horizontal component of the tensional force in the chain which causes Fido to be displaced to the right. The horizontal component is found by multiplying the force F by the cosine of the angle between F and d. This factor determines the portion of the force which actually causes a displacement.
Example #4: To pull Fido backwards a distance of 2.8 m, a force of 25 N is exerted on Fido's chain that is pulled at an angle of 45 to the horizontal motion. Find the work done.
Given: d = 2.8 m F = 25 N
= 45
W = ? Equation: W = Fd cos
Substitution: W = (25 N)(2.8 m)(cos 45) = (70 N*m)(0.707) Answer: = 49.49747468 N*m = 49 J
When determining the measure of the angle in the work equation, it is important to recognize that the angle has a precise definition - it is the angle between the force and the displacement vectors. Be sure to avoid mindlessly using any 'ole angle in the equation.
A B
Cabinet starts moving here
Question 5 Diagram
Energy Problems
(the answers below are based on significant
figures)
1. What is the kinetic energy of a 6 kg mass moving at 4 m/s?
2. A force of 800 N is needed to push a car across a lot. Two students push the car 35
meters. How much work is done?
3. A person lifts a 60 N package 3 meters off the floor.
a) How much work was done on the package?
b) What is the potential energy of the package?
4. A person carries a 58 kg package of books up a flight of stairs. The length along the stairs is 24 meters. The vertical height of the stairs is 12 meters.
a) How much work is done in raising the package? b) What is the change in the total energy of the package?
5. A man is moving a 500.0kg filing cabinet on a frictionless surface.
a. Suppose the man exerts a force of 5N for a distance of 1 m to get the cabinet moving and overcome the inertia of the cabinet, how much work did he do on the cabinet?
b. Next, you observe the cabinet that is already in motion, moving 3.5 meters horizontally from point A to point B, how much work did the man do on the cabinet during this interval?
6. You want to give a Joule to a special friend. What could you do?
7. A 2 gram bullet is fired from a pistol. It moves at 300 meters/sec. What is the kinetic energy of the bullet?
8. At breakfast a student ate Wheaties, the Breakfast of Champions. She noted on the box that each serving with milk would give her 50 Joules of energy. This student has no stored energy in her body at all. She wants to lift her empty breakfast bowl and take it to the sink. If she uses all 50 Joules to lift the bowl, how high can she lift it if the bowl has a mass of 500 grams?
9. An electron with a mass of 9.1 x 10-31 kg is moving at 2 x 108 m/s in the picture tube of a television set.
a) What is the kinetic energy of the electron?
b) How many of these electrons would it take to have 1 J of energy?
10. How many Joules of energy are needed to lift a fully loaded jetliner 10 km if the mass of the fully loaded jetliner is 7 x 104
kg?
*11. A penny has a mass of about 3.0 grams and is about 1.5 mm thick. You have 50 pennies stacked vertically -- piled on top of each other in a vertical stack.
a) What is the potential energy of the top penny in the stack? b) What is the potential energy of the entire stack of 50 pennies?
12. A 300 gram ball is thrown at you by a friend. The ball is moving very fast toward you but you are unable to quickly estimate the speed of the ball. You do know that the kinetic energy of the ball is 6000 Joules. What is the speed of the ball?
© Fyzics Softext, Inc. page 107
Answers:
1) 50 J 2) 3 x 104 J 3a) 200 J 3b) 200 J
4a) 6800 J 4b) 6800 J 5a) 5 J 5b) 0 J
7) 90 J 8) 10 m 9a) 2 x 10-14 J 9b) 5 x 1013 electrons
10) 7 x 109 J 11a) 2.2 x 10-3 J 11b) 5.5 x 10-2 J 12) 200 m/s
Energy Problems
1. A 2.1 gram bullet is shot into a tree stump. Just before it hits the stump the bullet is moving at 280 m/s. The bullet penetrates into the tree and comes to a rest after having gone 5.1 cm into the tree.
a) What was the initial kinetic energy of the bullet? b) What is the final kinetic energy of the bullet? c) How much work was done on the tree? d) What was the average force during the impact?
2. An 8.0 kg flowerpot falls from a window ledge 12.0 m above a sidewalk.
a) What is the kinetic energy of the pot just before it reaches the sidewalk? b) What is the speed of the pot just before it hits the walk?
3. A 300 kg iceboat is supported by the smooth surface of a frozen lake. The wind exerts a constant force of 400 N on the boat while the boat moves 900 m. Find the speed of the boat at the end of the run using the two different methods outlined below.
a) Use F = ma to find the acceleration of the boat. Then find the time it takes to move 900 m. Finally, find the final speed of the boat.
b) Find the final speed of the boat by equating the work done on it by the wind with the increase in kinetic energy.
4. A 17 kg model plane is flying at a horizontal speed of 14 m/s.
a) Find the kinetic energy.
b) The plane goes into a dive and levels off 21 m closer to the earth. What was the amount of potential energy lost in the dive?
c) How much kinetic energy was gained during the dive? d) What is the new kinetic energy?
e) What is the new velocity?
5. A rocket is loaded with fuel that gives it 1970 J of potential energy. The mass of the rocket is 9.5 kg. Assume that all the fuel is used to make the rocket accelerate.
a) How high will the rocket go?
b) What will be the maximum speed of the rocket?
6. A block weighing 67 N falls 64 meters.
a) What is the potential energy of the block before it falls? b) What is the speed of the block just before it hits?
*7. A 140 g baseball traveling 32 m/s moves a fielder's glove backwards 25 cm when the ball is caught. What is the average force exerted by the ball on the glove?
*8. If the speed of a car doubled, how much will the stopping distance increase by, assuming all other factors are the same? Ignore the driver's reaction time.
Answers:
1a). 82 J 1b). 0 J 1c). 82 J 1d). 1600 N
2a). 940 J 2b). 15 m/s
3a). 50 m/s 3b). 50 m/s
4a). 1700 J 4b). 3500 J 4c). 3500 J 4d). 5200 J 4e). 25 m/s
5a). 21 m 5b). 20 m/s
A
B
D E
C
Energy Problems
1. A skier with a mass of 65 kg comes down a slope moving at 15 m/s and proceeds horizontally off a cliff that is 22 m high. Find;
a) The initial kinetic energy of the skier.
b) The work done by gravity on the skier just before she hits the ground. c) The kinetic energy of the skier just before she hits the ground. d) The vertical speed of the skier just before she hits the ground. e) The velocity of the skier after the fall.
2. At a carnival a clown rides a unicycle and rides it down a slope as shown below. After the clown goes down the slope, the clown and cycle go through a loop where the entire clown-unicycle system will be upside down.
The height of the loop is 40.0 meters. The mass of the unicycle and the clown is 250.0 kg. The distance marked d on the diagram is 10.0 meters. Find;
a) The potential energy of the unicycle/clown in the beginning (point A).
b) The potential energy of the unicycle/clown at the bottom of the ramp, before entering the loop (point B). c) The kinetic energy of the unicycle/clown at the bottom of the ramp, just before entering the loop (point B). d) The potential energy of the unicycle at the top of the loop (point C).
e) The kinetic energy of the unicycle at the top of the loop (point C). f) The speed of the unicycle at point C.
g) The centripetal acceleration at point C.
h) Does the clown fall off the unicycle? Defend your answer with a numerical solution. i) What happens to the bowling pins the clown is juggling when he reaches point C?
3. A 50.0 kg skier starts at point “A” down the slope shown below moving at 10.0 m/s. Find the kinetic and potential energies and velocity of the skier at each point. (Point “A” is 20.0 m above the ground, “B” is 10.0 m, “C” is 8.00 m “D” is on the ground, and “E” is 2.00 m)
© Fyzics Softext, Inc. page 109
Answers:
1a). 7300 J 1b). 14,000 J 1c). 21,000 J 1d). 21 m/s
1e). 26 m/s, 54 below horizontal 2a). 120,000 J 2b). 0 J
2c). 120,000 J 2d). 98,000 J 2e). 25,000 J 2f). 14 m/s
2g). 10 m/s2 3A) EK = 2500 J; EP = 9800 J 3B) EK = 7400 J; EP = 4900 J; v = 17.2 m/s
3C) EK = 8380 J; EP = 3920 J; v = 18.3 m/s 3D) EK = 12,300 J; EP = 0 J; v = 22.2 m/s
Power
The quantity work has to do with a force causing a displacement. The amount of work does not depend on the amount of time that this force acts to cause the displacement. Sometimes, the work is done very quickly and other times the work is done rather slowly. For example, a rock climber takes an abnormally long time to elevate her body up a
few meters along the side of a cliff. On the other hand, a trail hiker (who selects the easier path up the mountain) might elevate her body a few meters in a short amount of time. The two people might do the same amount of work, yet the hiker does the work in considerably less time than the rock climber does. The rate at which a certain amount of work is done is known as the power. The hiker has a greater power rating than the rock climber does.
Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation.
Although the same amount of work is done by the rock climber and the hiker, the power
is different. In physics power is given the symbol P. The equation for power is P=W/t. In words, power is the rate of doing work. A powerful person can do work very fast. A less powerful person
may do the same amount of work, it will just take longer.
The metric unit for power will be work units divided by time units. Since work is in Joules, and time is in seconds, the power unit is Joules/sec. Because this unit is used so
frequently it is renamed a Watt. So power is in Watts.
Example #1: A horse jumps over a fence. It bends its knees and pushes against the ground with a force of 7100 N for 0.050 seconds. During this time the legs straighten out and the horse moves up 95 cm.
Given: F = 7100 N
t = 0.050 s
d = 95 cm = 0.95 m
P = ?
Equation: W=Fd and P=W/t
Substitution: W= (7100 N)(0.95m) = 6745 J
P= 6745 J/0.050 s Answer: = 134900 J/s = 130,000 Watts
From our original power equation, we can derive an equation that uses force and velocity.
P
=
W
t
=
Fd
t
=
F
∗
d
Power Problems
1. A light bulb is marked '60 Watts'. If the bulb is left on for one hour, how much energy will the bulb consume?
2. A mechanical system is being used to lift an elevator. The elevator weighs 8200 N. Two people step onto the elevator, each with a mass of 55 kg. They move up three flights (a total distance of 9.0 meters) in 6.0 seconds.
a) What is the total weight of the two people? b) What is the weight of the elevator with the people? c) What is the total work done in Joules?
d) What is the power of the mechanical system in watts?
3. A falcon (mass of 5.42 kg) is at rest in the branch of a tree 21.5 meters off the ground when it spots a happily Bouncing Bunny. It takes off horizontally, flapping its wings, doing a total of 4340 Joules of work in 10.0 seconds to get up speed. It then dives at its prey. If it takes 0.372 seconds to reach the bunny from the time it begins the dive.
a) What is the speed of the Falcon just before it strikes the Bouncy Bunny? b) What is the power that the Falcon used to get up to speed prior to diving?
4. Ram Bo is able to lift a 800.0 kg refrigerator 80.0 cm off the ground in 0.22 seconds. His friend, Nerd Berford, decides to show that he too can lift a refrigerator as well, but using some brains instead of just muscle. So Mr. Berford designs a ramp and pushes the refrigerator up the ramp to the same height as Mr. Bo. Now it takes Nerd a lot longer because he pushes up a little, then rests, then pushes and rests some more. In fact, it took 2.0 days, 3.0 hours, 17 minutes, 6.0 seconds for Nerd to make it to the top (At which time Ram called him Sisyphus or something like that). Find;
a) Ram Bo's power. b) Nerd Berford's power.
*5. The Colorado-Big Thompson Project is a diversion project that sends water from the Western Slope to the Eastern Plains. Water (270,000,000 m3/year) from Grand Lake flows down through a 13.1 mile tunnel to Carter Lake, Horsetooth
Reservoir and Boulder Reservoir. While the water drops, it generates an average of 515 million kWatt-hrs per year. Assuming perfect efficiency and no friction, how big of an altitude drop must there be in order for this to occur? (The density of water is 1000 kg/m3).
6. The Black Mountain Express quad chairlift at Arapahoe Basin can take a maximum of 2000 people per hour up the 220 m vertical rise of the mountain. If the average person has a mass of 50 kg (with ski/board equipment), what is the power needed in the lift engine?
© Fyzics Softext, Inc. page 111
Answers:
1) 220,000 J 2a). 1100 N 2b). 9300 N
Horsepower
The engineer James Watt invented the term horsepower. Watt lived from 1736 to 1819 and is most famous for his work on improving the performance of steam engines. We are also reminded of him every day when we talk about 60-watt light bulbs and kilowatts.
The story goes that Watt was working with ponies lifting coal at a coalmine, and he wanted a way to talk about the power available from one of these animals. He found that, on average, a mine pony could do 22,000 foot-pounds of work in a minute. He then increased that number by 50% and pegged the measurement of horsepower at 33,000 foot-pounds of work in one minute. It is that strange, arbitrary unit of measure that has made its way down through the centuries and now appears on your car, your lawn mower, your chain saw and even in some cases your vacuum cleaner!
What horsepower means is this. In Watt's judgment, one horse can do 33,000 foot-pounds of work every minute. So imagine a horse raising coal out of a coalmine as shown below. A horse exerting one horsepower can raise 330 pounds of coal 100 feet in a minute, or 33 pounds of coal 1000 feet in one minute, or 1,000 pounds 33 feet in one minute, etc. You can make up whatever combination of feet and pounds you like - as long as the product is 33,000 in
one minute and you have one horsepower. You can probably imagine that you would not want to load 33,000 pounds of coal in the bucket and ask the horse to move it one foot in a minute because the horse couldn't budge that big a load. You can probably also imagine that you would not want to put one pound of coal in the bucket and ask the horse to run 33,000 feet in one minute, since that translates into 375 miles per hour and most horses can't run that fast.
Horsepower can be converted into other units. For example, one horsepower is equivalent to 746 watts or 2,545 BTUs
(British Thermal Units) per hour. So if
you took a one-horsepower horse and put it on a treadmill, it could operate a generator producing a
continuous 746 watts. If you took that 746 watts and ran it through an electric heater, it would produce 2,545 BTUs in an hour (where a BTU is the amount of energy needed to raise the temperature of one pound of water one degree F). One BTU is
equal to 1,055 joules, or 252 gram calories, or 0.252 food Calories. Presumably the horse would burn 641 Calories in one hour doing its work if it were 100% efficient.
Horsepower Problems
1. The English unit for power is called a 'horsepower'. This was originally called a horsepower because it was the amount of work a standard workhorse could perform in a specified time period. Today the horsepower (hp) is defined as 1 hp = 746 Watts. On an electric saw the motor is known to develop 1.7 hp.
a) What is the power output of the saw in watts? b) How much work can the saw do in 10 minutes?
2. A mechanical system is being used to lift an elevator. The elevator weighs 8200 N. Two people step onto the elevator, each with a mass of 55 kg. They move up three flights (a total distance of 9.0 meters) in 6.0 seconds. If the power used is 14,000 watts what does the horsepower of the motor have to be?
3. An F-14 goes into a steep climb, going from an altitude of 2.1 km to 4.3 km in 24 seconds. The plane with the pilot has a mass of about 2.3 x 105 kg. What is the horsepower output of the engines?
*4. How long will it take a firefighter climb an 11 m vertical ladder while maintaining a power output of 0.25 hp? His mass with his equipment is 75 kg.
HORSE POWERS OF SOME CARS
CAR HORSE
POWER
Dodge Viper 450 Chevrolet Corvette 345 Porsche Carrera 300 Ford Escort 110
Simple Machines
You use machines every day. Typewriters, needles, bicycles, and doorknobs are machines.
Obviously, some machines are more complex than others. We can study the characteristics of machines by
analyzing simple machines, because all machines are modifications or combinations of six simple machines.
The six simple machines are the lever, the pulley, the wheel-and-axle, the inclined plane, the wedge, and the
screw. Consider the bottle opener. The opener does work by lifting the cap. The work you do is the input
work. The work the machine does is the output work.
The Simple
Machines We Will
Look At
What It Is
How It Helps Us
Work
Examples
LEVER A stiff bar that rests on a support called a fulcrum
Lifts or moves loads Shovel, nutcracker, seesaw, crowbar, elbow, tweezers, bottle opener
INCLINED PLANE A slanting surface connecting a lower level to a higher level
Things move up or down it
Slide, stairs, ramp, escalator, slope, wedge, screw
WHEEL AND AXLE A wheel with a rod, called an axle, through its center: both parts move together
Lifts or moves loads Car, wagon, doorknob, pencil sharpener, bike
PULLEY A grooved wheel with a rope or cable around it
Moves things up, down or across
Curtain rod, tow truck, mini-blind, flag pole, crane
Mechanical Advantage
Machines make-work easier to do even though they deliver less work than is supplied to them. One can understand how work is made easier by considering an ideal machine, which is a machine with 100% efficiency. In this machine
Win = Wout
work in = work out
The work in is the amount of work you do to operate the machine, while the work out is the amount of work needed to move the load in the machine. Or more precisely the input work is the product of the force that was exerted on the machine, called the effort force or the force in, Fin, and the displacement of this force, din . Likewise the output work is the product of the
force that the machine exerted, called the load force or the force out, Fout , and the displacement of this force, d out.
Since
W
in =F
ind
inand
W
out= F
outd
outand Wi = Wo
so Fin din = Fout dout
and so Fout/Fin = din/dout
Thus the ratio of Fout to Fin indicates the magnitude of the force of the load compared to the force of the effort. This ratio depends
upon the ratio of din to dout. By increasing the ratio of din to dout, the effect of the effort force can be multiplied. The left member
of the equation is called the actual mechanical advantage (AMA). The ratio of Fout to Fin indicates the amount that the machine
has increased the effect of the effort force.
AMA
=
F
outF
¿The right side of the equation is the ratio of din to dout and is called the ideal mechanical advantage (IMA)
IMA
=
d
¿d
outBy designing a machine to have a given ratio of din to dout, one can choose the ratio of Fout to Fin. In an ideal machine, the AMA
and IMA are equal. In real machines, the AMA is less than the IMA due to the work in overcoming friction.
The efficiency of any machine is the ratio of the output work to input work. Mathematically, the efficiency can be expressed as
eff = W
out/W
inx 100
In which eff is the efficiency, Wout is the output work, and Win is the input work. The answer is expressed as a percentage. For
an ideal machine, the efficiency is 100%; the output work equals the input work. In reality, the efficiency of a machine is less than 100%; the output work is less than the input work.
The efficiency of a machine can be expressed in terms of the AMA and the IMA. The efficiency of a machine is given by Eff = Wout/Win x 100
Eff = Fout dout/ Findin x 100 = Fout/Fin x dout/din x 100
However, since multiplying by dout/din is the same as dividing by din/dout,
Eff = AMA/IMA x 100
____________________________________________________________________________
Example: A delivery person moves a 215 kg refrigerator 0.75 m up three steps using a 4.0 m long inclined plane. The amount of force the delivery person uses to push the fridge up the ramp is 420 N. Find the efficiency of the inclined plane.
Givens: Fin = 420 N
Fout = mag = 215 (9.8) = 2107 N
din = 4.0 m
dout= 0.75 m
Equation: Eff = Wout/Win x 100 = Foutdout/Findin x 100
25 cm 75 cm
Simple Machine Problems
1. Write whether the following refer to din, Fin, Fout or dout: Load, effort, height, weight, applied force
2. If an object is moved 5.0 meters by a force of 3.0 N on the output side of a simple machine and a force of 2.0 N is applied for a distance of 8.0 meters on the input side of a simple machine:
a. What is the amount of work in? b. What is the amount of work out? c. What is the efficiency of the machine?
3. If an object is moved 10.0 meters by a force of 6.0 N on the output side of a simple machine and a force of 4.0 N is applied for a distance of 16 meters on the input side of a simple machine:
a. What is the ideal mechanical advantage of this machine? b. What is the actual mechanical advantage of this machine? c. What is the efficiency of this machine?
4. A force of 225 N is applied over a distance of 7.35 m so that a 1340-N weight can be raised to a height of 0.975 m. Find: AMA, IMA, Efficiency, Workin, and Workout.
5. A simple machine is 40% efficient and lifts an object weighing 15 N a distance of 2m high. What force will be used if the input force moves 10 m?
6. If work in equals work out, what is the advantage of using a simple machine?
*7. If an effort of 15 N is applied downward at the right end of the lever to lift the bucket, how heavy is the bucket (ideal situation)? What is the IMA of this compound machine? How high would the bucket rise if the effort pushed the right end of the lever down 30 cm?
© Fyzics Softext, Inc. page 115
Answers:
2a) 16 J 2b) 15 J 2c) 94%
3a) 1.6 3b) 1.5 3c) 94%
4) AMA=5.96 IMA= 7.54 efficiency =79.0 %
work in = 1650 J workout =1310 J
Energy, Power & Simple Machines Review Problems
1. A steel ball is dropped onto a thick steel plate. Its speed just before it hits is 12 m/s. What height was the ball dropped from?
2. A bungi jumper jumps off a vertical wall in Yosemite Valley. He falls 35 meters before the rope straightens out and begins to brake his fall. If he had been using a rope made of jute it would have only stretched 3.0 meters in stopping him, but this climber was using a nylon rope which stretched 7.0 meters before it stopped him.
a) What is the speed of the climber when the rope straightens out and just begins to break his fall? b) What would be the acceleration on the climber while stopping if he had been using a rope made of jute? c) What is the acceleration on the climber while stopping using the nylon rope?
d) What is the advantage of using the nylon rope?
3. A large car uses about 700. N of force just to overcome air resistance at a speed of 100. km/hr (about 60 miles/hr). If the car is traveling at a constant speed of 100. km/hr, how much horsepower is being used just to overcome air friction?
4. A cruel physics teacher requires her students to do sit-ups when they are tardy to class, one sit-up for every second tardy. (Great idea....!) A 63 kg student is 25 seconds tardy to class one day and has to do 25 sit-ups. She does these in 16 seconds. What is the power output of the student in horsepower for that time? (You may assume that half of the students mass is lifted for each sit-up and that the mass is lifted 38 cm off the ground).
5. An electric motor develops 65 kilowatts of power as it lifts an elevator 18 m in 40.0 seconds. How much force does the motor deliver?
6. A 10.0-gram ball is shot straight down at the floor from 1.2 meters above the floor. When it hits the floor the ball is moving at 6.7 m/s. What was the initial speed of the ball?
7. A shot-putter pushes a 7.0 kg shot from rest to 13 m/s in 1.7 seconds.
a) How much work did he do?
b) What was the power required in watts?
8. When the bill comes from XCel Energy you are charged for electricity in a unit called a kilowatt-hour (kilowatts times hours). This is a billing for energy. If a home uses 650 kW-hr of electricity in one month, what was the energy consumption in Joules?
9. During a single beat of a female's heart about 50 grams of blood is pushed upward at a constant speed of 1.2 m/s. What is the power output in Watts of a female heart?
10. A 400-N load is lifted 1 meter using an inclined plane. The inclined plane is 5 meters long. The effort used to push the load up the plane is 100 N. Find: AMA, IMA, Efficiency, Workin, and Workout.
11. A force of 160 N is used to lift a 600 N load to a height of 5 meters using a pulley system. The 160-N force is used to pull a rope through a distance of 25 meters. Find: AMA, IMA, Efficiency, Workin, and Workout.
12. A machine is 75% efficient. How much work must be put into the machine to get 60 J of work out of the machine? What causes a machine to be less than 100% efficient?
13. A 4.5-meter plank is used as a ramp to raise a 965-N box to a height of 1.68 meters. The effort required to push the box up the ramp is 455 N. Find: AMA, IMA, Efficiency, Workin, and Workout.
14. A pulley system with 4 supporting strings lifts a 1.000-kg mass with an effort of 3.4 N. Find: AMA, IMA, and Efficiency. How far must the effort be applied to lift the mass to a height of 12 cm?
16. A force of 225 N is applied over a distance of 7.35 m so that a 1340-N weight can be raised to a height of 0.975 m. Find: AMA, IMA, Efficiency, Workin, and Workout.
17. A student rotates the handle of a manual pencil sharpener (a wheel and axle). The radius of the inner rod that is connected to the sharpening screws is 0.51 cm. The length of the handle is 5.2 cm. If a force of 4.3 N is applied to the handle, what is the force that the inner rod can exert on the sharpening screws? What is the IMA of this part of the machine?
18. A 9800-N generator is to be loaded onto a truck that is 1.5 meters above the ground. Find the length of the shortest ramp that could be used if the maximum effort applied to the generator is 440 N for an ideal situation.
*19. A 2.0-meter crowbar is used to lift a 1750-N stone from the ground. The fulcrum is 25 cm from where the stone rests on the bar. What is the IMA of the crowbar? What force must be exerted at the other end of the bar to move the stone if the machine is ideal? What force must be exerted if the machine has an efficiency of 85%? If the input force acts through a distance of 45 cm, how far does the stone rise?
© Fyzics Softext, Inc. page 117
Answers:
1) 7.3 m 2a) 26 m/s 2b) –110 m/s2 2c) –48 m/s2
3) 26.1 hp 4) 0.25 hp 5) 140,000 N 6) 4.6 m/s
7a) 590 J 7b) 350 W 8) 2.3E9 J 9) 0.6 W
10) 4 5 80% 500 J 400 J
11) 3.75 5 75 % 4000 J 3000 J
12) 80 J
13) 2.1 2.7 78 % 2000 J 1620 J
14) 2.9 4 72 % 48 cm
*15) 0.16 m
16) 6 7.5 80 % 1650 J 1310 J
17) 44N 10 18) 33 m
Answer(s)
Terms
Choices for Units
1. AMA
a)
horsepower2. Potential energy
b)
kg3. Force
c)
m/s4. IMA
d)
m/s25. Momentum
e)
N6. Speed
f)
J7. Kinetic Energy
g)
watts8. Power
h)
Ns9. Weight
i)
no units10. Acceleration
j)
Percentage 11. Efficency12. Mass 13. Work
Answer(s)
Questions
Choices for Answers
14. Which situation to the right would have the greatest
gravitational potential energy? a) A bowling ball falling at 2.2 m/s but is still 4 m above the ground 15. Which situation to the right would have the greatest kinetic
energy?
b) A bowling ball not moving 1.5 m above the ground.
16. Which situation to the right would have the greatest
total energy? c) A bowling ball falling at 6.2 m/s just above the ground 17. Which situation to the right would have the least
kinetic energy? d) A bowling ball falling at 7.2 m/s but is still 5 m above the ground 18. Which situation to the right would have the least
potential energy?
19. If a person pushes against an unmoving wall with a force of 16N why is there no work being done?
20. What type of energy does an object have as the object falls from the roof of a building but it is only half way to the ground?
21. What type of energy does an object have if it fell from the roof of a building and it is just hitting the ground?
22. What is it called if an object is lifted to the roof of a building?
23. What is it called if an object has only stored energy?
24. What is done to a tree stump if a bullet is fired into the tree stump and the stump stops the bullet?
25. What type of energy does an object have if it is moving on the ground?
26. If a 5kg object is moving on a frictionless surface at a constant speed of 2m/s how much work is being done?
27. If a hot wheels car goes down a ramp and around a loop-the-loop without falling what must be true of its centripetal acceleration?
29. True or false using a simple machine increases the energy you put into the machine.
30. True or false using a simple machine allows you to lift a large load with less effort by moving the effort a large distance.
31. True or false using a simple machine reduces the amount of work needed to do a task.
32. True or false a person running up some steps does more work compared to if they ran up the steps.
33. True or false a person running up some steps needs more power compared to if they walked up the steps.
37) If a 20.0 kg bowling ball 10.0 m up in the air is moving downward with an initial velocity of 10.0 m/s. a) What is its potential energy when it is 10m up?
b) What is its kinetic energy when it is 10m up?
c) What is its total energy when it is 10m up?
d) What is its kinetic energy a it strikes the ground
e) What is its velocity a it strikes the ground
© Fyzics Softext, Inc. page 119
Work:
show all work to Find the spring scale reading for the pulley system.Pulley system
34)
Load = 10 kg
1
0
kg
Spring Scale
Use a double pulley here
35)
Spring Scale
10
kg Load = 10 kg
36)
Load = 10 kg
Spring Scale1
0k
41) A 250 kg rock starting from rest is pushed across a frictionless ice covered lake with a force of 695N over a distance of 25 m.
a)What will be the acceleration of the rock?
b) How fast will the rock be going at the end of the 25 m?
42) How much does a refrigerator weigh if it takes 70 N of force to push the refrigerator up a 22m long ramp, causing the refrigerator to be 1.2 m up in the air?
43) What is the efficiency of a machine that lifts a 1100 N refrigerator 1.2 m into the air using a force of 70N along an inclined plane that is 22m long.
Work
show all work Lever System38) Find the effort for the lever system to the left.
d = 10m
Eff
o
rt
L
oa
d
1
0
k
g
d = 5m
For #37
39) Find the distane in for the lever system to the right.
d = 10m
Eff
o
rt
3
k
g
L
oa
d
1
0
k
g
d = ?
For #38
40) Find the effort for the lever system to the left.
d = 10m
Eff
o
rt
L
oa
d
2
0
k
g
d = 15m
17.5 m
4.3 m
A
B
44) What is the power in watts needed to lift a 555kg door 2m into the air in 3 seconds?
45) What is the Horse power needed to lift a 555kg door 2m into the air in 3 seconds?
46) A person carries a 56 kg package of books up a flight of stairs. The length along the stairs is 34 meters. The vertical height of the stairs is 18 meters. How much work is done in raising the package?
47) An unmoving Skier with a mass of 50.0 kg, at point A skies down the frictionless slope. What is the potential energy, kinetic energy and velocity of the skier at point B?
© Fyzics Softext, Inc. page 121
Answers:
1) i 2) f 3) e 4) i 5) h 6) c 7) f 8) a & g 9) e 10) d 11) j 12) b 13) f 14) d 15) d
16) d 17) b 18) c 19) It is unmoving 20) Kinetic & Potential
21) kinetic 22) work 23) potential 24) work 25) kinetic
26) 0J no force is needed to keep it moving 27) greater than or equal to 9.8m/s2
28) False 29) False 30) True 31) False 32) False 33) True
34) 24.5N 35) 49N 36) 98N
40a) 1960 J b) 1000 J c) 2960 J d) 2960 J e) 17 m/s
37) 20kg or 196N 38) 33m 39) 13kg or 131N
41a) 2.8 m/s2 41b) 11.8 m/s 42) 1283 N 43) 86%
44) 3626 watts 45) 4.9 hp 46) 9900J
