Download 1 Unit 3 Momentum and Energy In this unit we are going to be

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Unit 3
Momentum and Energy
In this unit we are going to be looking at the two most fundamental laws
of nature – the law of conservation of energy and the law of
conservation of momentum.
Chapter 6 – Work, Power and Efficiency
Every object you see has some form of energy. There are two types of
energy an object can have – kinetic energy and potential energy.
Kinetic energy is the energy of motion and potential energy is the energy
that is stored in an object. Mechanical energy is the combination of
both kinetic and potential energy.
What is Work?
Work is not energy itself but it is a transfer of mechanical energy. A
force does work on an object if it causes the object to move. Note that
work is always done on an object and results in a change in the object.
Physics defines work very specifically as force acting on an object
causing the object to move a certain distance. The amount of work
depends directly on the magnitude of the force and the displacement of
the object along the line of the force. Work is the product of force and
displacement when the force and displacement vectors are parallel and
pointing in the same direction. W = F║Δd (W is scalar so vector
notations are not used.)
What is Not Work?
Because of the specific definition of what work is there are several
situations that may appear to involve work but from a physics point of
view, actually involve no work.
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Case 1: Applying a force that does not cause motion. The most
common example for this situation is looking at a student pushing on the
classroom wall. He may appear to be doing work. He feels tired after a
while and has been exerting a force (expending energy) upon the wall.
But, because the wall didn’t move or experience a change in
displacement, no work was done.
Case 2: Uniform motion in the absence of a force. This ties into
Newton’s first law – an object will continue in motion unless acted upon
by an external force. If we look at a hockey puck sliding along a
frictionless surface, there was work done to get the puck moving but
there is no work being done to keep it moving. Since the puck is not
changing its motion, no force is acting upon it.
Case 3: Applying a force that is perpendicular to the motion. If you
look at carrying your physics book down the hall, both you and the book
are moving. The force acting on the book however is perpendicular to
the line of motion since your arm is exerting a force up. Picking up the
book is work forces are parallel. Once you and the book are moving at a
constant velocity, you are no longer doing work on the book. Another
example of perpendicular forces is swinging an object on a string.
Calculating Work
The amount of work depends directly on the magnitude of the force and
the displacement of the object along the line of the force. Work is
calculated using the equation: W  Fll d W is the work done, d is the
displacement of the object and F is the force exerted on the object.
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The derived unit for work is the joule, J, and is named after nineteenth
century physicist, James Prescott Joule. One joule is equivalent to a
N m
Practice Question
A student is rearranging her room. She decided to move her desk across
the room, a total distance of 3.00 m. She moves the desk at a constant
velocity by exerting a horizontal force of 200 N. Calculate the amount
of work the student did on the desk in moving it across the room.
Page 221 #1, 2, 3; Page 225 #4 – 10
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Constant Force at an Angle
It is rare in everyday situations that the forces acting on an object will be
precisely parallel or perpendicular to the motion of an object. If we look
at a situation similar to that in figure 6.11 on page 230, the applied force
is at an angle relative to the direction of the wagon’s displacement. To
determine the work being done by the child pulling the wagon, you must
first find the component of the force that is parallel to the direction of
motion.
This will be done in a very similar way to how we found vertical and
horizontal components of forces in the last unit. We will use
trigonometry to resolve force into it components.
When the force and displacement vectors are not parallel and are
pointing in the same direction, work can be calculated using the formula:
W  Fd cos .
According to this equation, the work being done by gravity will be 0 J.
Gravity will being acting at 90o to the direction of displacement. The
cosine of 90o is 0 so the work being done will also be 0.
Practice Question:
How much work is being done on a wagon being pulled with 80 N of
force at an angle of 23o to the direction of motion?
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Positive and Negative Work
Positive work adds energy to an object; negative work removes energy
from an object. If we look at the common force of friction, we can see
how our new equation for work, W  Fd cos , will show that friction
results in negative work.
If we were to drag a block across a desk, friction is going to work in the
direction opposite to the direction of displacement. Friction can be said
to be acting at 180o to the direction of displacement. Because the cosine
of 180o is -1, the resulting work being done by friction will be negative.
Practice Problem:
A weight lifter is bench-pressing a barbell weighing 650 N through a
height of 0.55 m. There are two distinct motions: (1) when the barbell is
lifted up and (2) when the barbell is lowered back down. Calculate the
work that the weight lifer does on the barbell during each of the two
motions.
Page 235 #14 – 18
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Kinetic Energy
The energy of motion is called kinetic energy. If you think about a few
simple situations, you can begin to understand what quantities will be
involved in calculating kinetic energy.
If there was a bowling ball and a golf ball rolling toward you at the same
velocity, which would you try to avoid? If there were two golf balls of
the same mass coming toward you, one slow and one fast, which would
you try to avoid?
Your answers to these questions should lead you to think that mass and
velocity play a role in the amount of kinetic energy an object possesses.
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The equation for kinetic energy is: E k  mv
Kinetic energy is measured in joules and is a scalar quantity.
Practice Question:
A 200 g hockey puck, initially at rest, has a final velocity of 27 m/s.
Calculate the kinetic energy of the hockey puck a) at rest and b) in
motion.
Page 238 #19 – 21
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Work and Kinetic Energy
The special relationship between doing work on an object and the
resulting kinetic energy of the object is called the work-kinetic energy
theorem. Because doing work on an object results in the object having
velocity, there is kinetic energy.
Before we make a mathematical expression for this theorem, we must
make one assumption. That is to assume that all work done on a system
results in kinetic energy only. Page 240 shows the entire derivation.
The equation for the work – kinetic energy theorem is
W  Ek or W  Ek 2  Ek1
These representations describe how doing work on an object can change
the object’s kinetic energy (energy of motion). The work-kinetic energy
theorem is part of the broader work-energy theorem that includes the
concept that work can change an object potential energy, thermal
energy, or other forms of energy.
Practice: A physic student does work on a 2.5 kg curling stone by
exerting 40 N of force horizontally over a distance of 1.5 m.
a) Calculate the work done by the student on the curling stone.
b) Assuming that the stone started from rest, calculate the velocity
of the stone at the point of release. (Consider the ice to be
frictionless)
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Practice 2: A 75 kg skateboarder (including the board), initially moving
at 8.0 m/s, exerts an average force of 200 N by pushing on the ground,
over a distance of 5.0 m. Find the new kinetic energy of the
skateboarder if the trip is completely horizontal.
Page 245 – 246 #22 – 26
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Potential Energy and Work-Energy Theorem
It is possible to do work on an object but the object does not gain kinetic
energy. Doing work on an object can result in a change in potential
energy rather than kinetic energy. Potential energy is the energy stored
by an object due to its position or condition.
By doing work against the force of gravity a special form of potential
energy called gravitational potential energy is gained. This is only one
of several forms of potential energy. Chemical potential energy is stored
in the food we eat, elastic potential energy is stored in elastic bands, and
batteries store electrical potential energy.
Gravitational potential energy has been used for hundreds of years to
produce energy from water sources starting with water wheels and now
includes reservoirs and dams.
There are several factors that play a role in calculating gravitational
potential energy. If you think about a Ping Pong ball and a golf ball
dropped from the same height, which would you try to catch? What
about a golf ball dropped from 10 cm and one dropped from 10 m,
which would hit the ground harder? Would a golf ball dropped from 1
m hit the surface of the Earth or moon harder?
Your answers to these questions should tell you that gravitational
potential energy is affected by mass, height and the acceleration due to
gravity.
An important characteristic of all forms of potential energy is that there
is no absolute zero position. You must assign a reference position and
compare the potential energy of an object to that position. Typically the
reference position is the solid surface toward which an object is falling
or might fall.
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The equation for gravitational potential energy is E g  mgh .
In this equation, h is the change in height from the reference position.
Practice Question: You are about to drop a 3.0 kg rock onto a tent peg.
Calculate the gravitational potential energy of the rock after you lift it to
a height of 0.68 m above the tent peg.
Page 250 #27 – 29
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Work and Gravitational Potential Energy
Similarly to work and kinetic energy, the equation for the work-energy
theorem in terms of gravitational potential energy can be given by:
W  Eg or W  Eg 2  Eg1 . The derivation for this formula can be found
on page 251.
Practice Problem: A 65.0 kg rock climber did 1.6 x 104 J of work against
gravity to reach a ledge. How high did the rock climber ascend?
Page 254 #30 – 34
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Elastic Potential Energy
When an object can stretch, compress, bend, or change in shape in some
way and then return to its original position, it is said to be elastic. The
energy stored in an object with elastic properties has a form of stored
energy called elastic potential energy.
Hooke’s Law
Hooke’s law says that the force applied to extend or compress a spring
will be proportional to the amount of extension or compression of the
spring. The restoring force, the force exerted by the spring to return it
to its original position, will act in the opposite direction to the applied
force. Hooke’s law is given by the equation F = -kx for the restoring
force and Fa = kx for the applied force.
In this equation, x is the amount of extension or compression of the
spring and k is the spring constant. Each spring has a different spring
constant that is measured in N/m.
Practice Question: A typical archery bow requires a force of 133 N to
hold an arrow at “full draw” (pulled back 71 cm). Assuming that the
bow obeys Hooke’s law, what is its spring constant?
Page 258 #35 – 37
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Work and Elastic Potential Energy
Elastic potential energy for a perfectly elastic material is given by the
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equation: E  kx
It is important to note that there is no material that is perfectly elastic.
For a material to be perfectly elastic, it will return to precisely to its
original form after being deformed. All materials will reach an elastic
limit where they will not return to their original shape if stretched to that
limit.
Practice Question: A spring with a spring constant of 75 N/m is resting
on a table.
a) If the spring is compressed a distance of 28 cm, what is the
increase in its potential energy?
b) What force must be applied to hold the spring in this position?
Page 261 #38, 39, 40
END for Jan 2010
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Power
Power is the rate at which work is done. It is the rate at which energy is
transferred. It is given by the equation P 
W
E
or P 
t
t
The unit for power is the watt, W. Remember that work is done on an
object and results in a transfer of energy to that object.
Practice Question:
1. A crane is capable of doing 1.50 x 105J of work in 10.0 s. What is
the power of the crane?
2. A cyclist and her mountain bike have a combined mass of 60.0 kg.
She is able to cycle up a hill that changes her altitude by 4.00 x 102
m in 1.00 min. (Assume that friction is negligible.)
a. How much work does she do against gravity in climbing the
hill?
b. How much power is she able to generate?
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Efficiency
It is inevitable that some energy will be lost when transferring from one
type to another, usually in the form of heat. The efficiency of a machine
or device is a ratio of the useful energy to the total energy. It is
E
calculated using the equation: Efficiency  o  100% or Efficiency  Wo 100%
EI
WI
Eo and Wo are the useful output energy and work, respectively and EI
and WI are the energy and work inputted.
Practice Question: A model rocket engine contains explosives storing
3.50 X 105 J of chemical energy. The stored chemical energy is
transformed into gravitational potential energy at the top of the rocket’s
flight path. Calculate how efficient the rocket transforms stored
chemical energy into gravitational potential energy if the 500 g rocket is
propelled to a height of 100 m.
Page 266 #41, 42, 43
Page 270 – 271 #44 – 50
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Work Done When Forces are Changing
Much like in our study of kinematics, there are often situations where
the forces are not uniform. To determine the work done when the force
acting upon an object is not uniform, calculating the area under the curve
of a force – position graph will give the work done.
Often the graphs we examine will be fairly straightforward similar to the
example given on page 227. If you happen to encounter a graph that is
not so easy to work with (see figure 6.10 on page 226), you must count
all of the squares under the curve and estimate for partial squares.
Although this method is not completely accurate, the mathematical
solution for such a problem will not be taught in this course.
Complete the example on page 227.
Page 229 #11, 12
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