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Kinetic and gravitational potential energy.
Let's consider a dropped ball again.
As I release it, its velocity is zero.
As it gets to the floor, it's attained a finite velocity.
From what we know already, we can figure out
the relationship between the height h from which
it was dropped and the speed v, which it had when
it hit the floor:
distance fallen h = average speed x time=(1/2)vxtime
v=time x g
Therefore time =v/g. Put it in the 1st equation:
h=(1/2)v(v/g) or gh=(1/2)v2
Kinetic and gravitational potential energy.
From F=ma and the fact that the gravitational
force is mg, we thus conclude that, for the dropped ball:
gh=(1/2)v2
or multiplying both sides by m
mgh=(1/2)mv2
at start just before it hits
We say that the ball had
Gravitational potential energy =mgh
At the start of its fall and that energy changed
form and became
Kinetic energy = (1/2)mv2
Just before the ball hit the floor
Conservation of Energy
In fact, a more general analysis shows that
at any moment during the fall, the sum
Gravitational energy + kinetic energy
stays the same. At the beginning of the fall,
the energy is all gravitational. At the bottom
of the fall it's all kinetic. In between, a gradual
change of form of the energy takes place, but the
total amount of energy stays the same.
This is what happens to the energy of water going over
falls or a dam. In hydropower systems, the energy
is stored in the water above the dam, and then converted
to kinetic energy as it goes over the dam. The kinetic
energy in the water is then used to do useful things
like generate electricity as we will discuss later.
Hoover Dam on
the Colorado river
In Nevada
Height of the
water at the
top above the
bottom= 180m
Volume of water in
The lake above the
Dam= 32 km3
Review
velocity is (change in position)/time elapsed
acceleration is (change in velocity/time elapsed
total force =mass x acceleration
gravitational force = Mass x g down
From F=ma for falling object
(1/2)mv2 + mgh = constant (conservation of energy)
kinetic
gravitational
potential
If I dropped a ball from a height of 2 meters,
what was its speed just before it hit the floor?
A. 9.8x2 m/s
B. 9.8/2 m/s
_____
C. √9.8x2 m/s
_____
D. √9.8x4 m/s
E. 9.8x4 m/s
Units of force:
Since total force is mass x acceleration
a suitable metric unit is
1kg meter/second2 called 1 Newton
In the British system, the force unit chosen is
the pound. The British mass unit (not used
much) is a slug and
1 pound = 1slug ft/second2
1 Newton = .225 pounds
We will tell you this conversion value if you
need it. (Not in Appendix B).
Weight:
The magnitude of the gravitational force on an
object is called its weight.
Therefore the weight of an object of mass M
is always Mg near the surface of the earth.
Because g varies with altitude, the weight can
be different in different places for a given object
but the mass does not change.
The units of weight are therefore force units.
Units of energy:
We can figure out the units of energy
from the expression for the gravitational
potential energy:
mgh= (a force) times (a length)
So metric units of force are Newtonsxmeters
which we call joules.
1 joule = 1 newton meter
British units of energy can then be
1 ft-pound
Confusingly, this is NOT a British Thermal Unit (btu)
1Btu= 1055Joules=778 ft-lb
Again, you do not have to remember the conversion
factors which are listed in Chapter 3, Appendix
B and inside the back cover of your book.
However you are expected to be able to use the
conversion factors to change a quantity from
one kind of units to another.
It is also important to always use the right
KIND of units: energy units for energy, force units
for force, etc.
Energy transfers between objects.
Consider holding up the book again.
Suppose its mass is 1kg and that I
slowly raise it 1 meter. How much
gravitational potential energy did it gain?
A. 1 joule
B. 9.8 joules
C. 1 newton
D. 9.8 newtons
OK it gained 9.8 joules. Since energy is
conserved, where did this energy
come from?
A. The kinetic energy of the book.
B. chemical energy in my body
C. the action of the friction of the air.
D. chemical energy in the book.
Now I want to concentrate on how the
energy was transferred from my body
to the book.
I pressed my hand against the bottom
of the book, and produced an upward
force which was just slightly larger than
the downward gravitational
force mg so that the book slowly
moved up a distance h. In magnitude the
resulting gain in energy was
mg x h which is equal to
(the upward force with which I pushed
with my hand)
times
(the distance through which I pushed the
book along the direction of the force. )
This is an example of the transfer of
energy from one object to another
through the performance of WORK.
When one object exerts a force F on
another and the second object
moves along the direction of
the force a distance d, then we say that
the first object has done work Fd on
the second object, resulting in energy
transfer Fd from the first to the second object.
Notes about work:
Work has the units of energy but it is always
an amount of energy transferred from
one object to another and NOT the amount
of energy in a object. After the energy
is transferred through the performance of
work, the energy ends up in the second
object in one of the forms we will be discussing
such as gravitational potential, kinetic,
thermal, chemical or electrical potential energy.
In the formula work=Fd, d is the distance
ALONG THE FORCE. This may not be the same
as the distance which the contact point between
one object and the other moves.
Suppose you slowly carry an object weighing
10 pounds up a flight of stairs consisting of
20 steps. Each step is 1/2 feet high and 1 foot
wide. How much work did you perform on the
object?
A. 100 ft-pounds
B. 200 ft-pounds
C. (5)1/2x100 ft-pounds
D. 0
You carry the same 10 pound object
100 ft down the horizontal hall.
During the time between the time
after you picked it up and before you
put it down, how much work did you
perform on the object? (Suppose
you walked slowly at constant
velocity).
A. 1000 ft-pounds
B. 0
C. 500 ft-pounds
D. insufficient information is given
From a zero velocity start, I push a cart
along a frictionless track with a constant
force for 1/2 meter and then let it go.
Its velocity after I release it is observed
to be 1m/s and the mass of the cart is
1/4 kg. What was the magnitude of the
force with which I pushed it?
A. 0.25N
B. 2.45N
C. 0.5N
D. 0
Question 6
I throw a 1/2 kg object straight up.
During the throw, I exert a constant
force upward on the object while it
moves upward through 1/2 meter.
Just after I release the object, it is observed
to have an upward velocity of magnitude
(speed) 2 meters per second. What was the
magnitude of the force which I exerted on the
object during the throw?
A. 4.8 N
B. 6.9 N
C. 0.25N
D. 0.5N
Question 7
Answer: B
The point here is that the object ends up with
two forms of additional energy, kinetic and
gravitational potential:
Fd
Work
Done
=
(1/2)mv2 +mgd
gain in
kinetic
energy
gain in gravitational
potential energy
F(1/2 m)=(1/2)(1/2kg)(2m/s)2 +(1/2kg)(9.8m/s2)(1/2 m)
Divide both sides by ½ meter and simplify
F=2kgm/s2 + 4.9 kgm/s2 = 6.9 newtons
OK that was the force exerted by
my hand on the ball. What was the
total force on the ball during the toss?
(1/2 kg ball, pushed up 1/2m, final
speed 2 m/s).
A. 6.9 newtons
B. 4.9 newtons
C. 2 newtons
D. 0
Question 11
Summary:
Energy is transferred from one object to another
through the performance of work. The amount
of work done by one object on another is
Work done = F x d
where F is the force which the first object exerts
on the second and d is the distance, along the
direction of the force, through which the force
acts. The work done is the amount of energy
which is transferred from the first object to
the second. The energy ends up in one of the forms
we are discussing: kinetic, gravitational potential,
thermal. It is not meaningful to talk of work IN a
body. If the force acting is opposite in direction
to the direction through which it acts, the work
done is negative and energy is extracted from
the second object.
Work is done both ways. Newton’s third law.
Newton’s third law concerns the relationship
between the forces which objects exert on each
other. Consider the book. As I hold it, I exert
an upward force of Mg on the book
(if it isn’t accelerating) and the book
exerts a force back on my hand. Newton’s third
law states that this force which the book exerts back
on my hand is equal in magnitude and opposite in
direction to the force which my hand exerts on the book.
Notice that these two forces are acting on different
objects. In this case they are both electromagnetic.
They have magnitude Mg only if the book is not
accelerating
Now how much work did the book do on my hand
as I raised it a distance d, assuming that I raised
It very slowly.
a.
b.
c.
d.
A, Mgd
B. -Mgd
C. 0
D. Insufficient information given.
Question 8
Answer is B: There are two ways to
see this:
1. Since energy is conserved in
the transfer process:
Mgd was transferred from my hand to the
book.
Therefore -Mgd of energy was transferred from
the book to my hand. The work done is the
energy transferred so the work done by
the book is –Mgd
2. The force with which the book acts on my
hand is down and that force acts through a
distance d up. So the product is
Work done by book=(F)d=-Mgd
I.
III.
a
II.
Consider the three forces:
I.The force of gravity acting down on the book, mass M.
II. The force of my hand on the book, pushing up.
III. The force of the book pushing down on my hand.
If the book is accelerating upward which of the following
is true?
A. Forces I,II and III all have magnitude Mg.
B. Force I has magnitude Mg. II and III have the same
magnitude not equal to Mg.
C. Force I has magnitude Mg. II and III are different from
Mg and from each other.
D. None of the forces has magnitude Mg when the
book is accelerating.
Question 9
It’s B. II and II have the same magnitude
by Newton’s 3rd law, but II is greater in
magnitude than Mg because there is
an upward acceleration.
Power
In many applications of energy transfer, the question
Of HOW FAST the energy is transferred is important.
The rate of energy transfer is measured by the power:
Power=
(amount of energy transferred)/(time required for the transfer)
The metric unit of power is 1 joule/second which is
called a watt:
1 joule/second = 1 watt
British units of power include 1ft-pound/sec
and the horsepower
1 horsepower= 746 watts
Thermal Energy:
In the energy economy we will be studying, thermal
energy plays a central role, as we have already
mentioned several times.
You are familiar with measurements of temperature
which are related to the amount of thermal energy
in an object.
However we will see that temperature does not directly
give us the amount of thermal energy in an object
or substance.
Thermal energy is the kinetic and potential energy
associated with the random motion of the atoms inside
a material that is heated.
There are two ideas here:
1. All ordinary matter is made of atoms. (In extraordinary
environments, such as the interior of the sun, the
atoms are broken up into constituent parts.) You
have probably heard this and we will return to it. You
do not have to know much about atoms to understand
thermal energy though, and the essential features were
understood before the atomic nature of matter was
established. Atoms have approximate size 10-10
meters and cannot be seen with our eyes.
Scanning Tunneling Microscope image
of the surface of a silicon crystal. The atoms
are between 10-10 and 10-9 meters apart.
Images like this have only been possible to
make for about 25 years.
If the atoms are about 10-10 meters apart,
about how many atoms are in a glass of
water approximately 0.1 m tall and 0.1m
in diameter. (Hint: you can make a rough
estimate which is good enough for our
purposes here by assuming that the glass
is a cube 0.1m on a side.)
A. 1010 atoms
B. 1011 atoms
C. 1013 atoms
D. 1027 atoms
E.
1030 atoms
Question 12
2. A kind of potential energy associated with the forces
between atoms is involved in thermal energy. This can be
illustrated with a spring between two balls. As I push the
balls together I do work and transfer energy to them which
is stored as potential energy (of electromagnetic origin).
If the temperature of
an object increases, its thermal energy increases.
One can measure how much it increases if
you assume that energy is conserved in
a process, such as the one you are doing in
the lab this week, in which mechanical
energy is converted to thermal energy.
In the simplest version of the experiment,
you rub one object against another. Unlike
our previous experiments, we arrange it so
that there is a lot of friction between the surfaces.
you know from experience that rubbing two
surfaces like that together causes heating.
thermometer
Fext
Ff
Ff
The block is pusheddwith force Fext
which could, for example come from
my hand. A frictional force Ff acts to the
left and exactly balances the external force
so that the block moves at steady speed.
Fext does work Fextd. Where does the energy
transferred go?
a. A. Kinetic energy of the block.
b. B. Gravitational energy of the block
Question 10
c. C. Thermal energy
d. D. Kinetic energy of the surface over which block is pushed
Since there was no acceleration, it must have ended
up as thermal energy. We will suppose (as
can be arranged) that the thermal energy
goes primarily into the block and very little into
the underlying surface. Now imagine doing
this experiment several times. Each time, after
pushing the block a distance d, read the thermometer
to find out how much the temperature rises. (You
would have to wait a while after pushing for the
temperature reading to stabilize and you would
have to wait longer for the temperature to go down
to room temperature before pushing the block again.)
You could plot up the data taken as
Temperature rise versus work done Fextd
Temperature rise
Work done=Fextd
In the cases we will consider, the graph of
temperature done versus work done will
be a straight line. The SPECIFIC HEAT
of the material in the block is defined as
Specific heat = (thermal energy rise)/(Mass x temperature rise)