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1206 - Concepts in
Physics
Monday, September 28th
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Notes
Tutorial this afternoon: we will go through the
solutions of assignment 1 in detail and than have
time for questions.
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New assignment posted on webpage under today,
due on Friday. If this is a serious problem for you,
please come and see me after class.
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We will use the tutorial time on Monday,
November 02nd - 16:30 to 18:00. If you have a
conflict, please come an see me before next
Monday.
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A selection of exercises that go with each lecture
from the text book will be posted on the
webpage. They are meant for those who want
more practice. In general all execises in the
textbook are suitable.
Notes for assignments
•
Note what is given and what you are looking for.
Problem: not clear what was asked.
•
Make a sketch/drawing where appropriate. This will
give you points already.
•
•
Make notes what you are doing.
•
Units are important, make sure you carry them
through. Stay with the formula as long as possible,
putting in numbers is the last step.
•
Make sure you copy correctly from the sheet.
Write out the formula or theorem you are using.
Easier (or possible) to follow your thought process
and see where it went wrong.
Units
• We usually give our results in SI units
• Sometime it is not straight forward to see
how we get to the final unit.
Reminder: 1 J = 1 N m = 1 kg m/s2 m = 1 kg/s2
Let’s look at potential energy:
PEgrav = m*g*h
kg *
2
m/s
now units on the right side:
* m = kg
2
m/s
*m=N*m=J
Let’s look at kinetic energy:
KE = 1/2 m * v2
now units on the right side:
kg * (m/s)2 = kg m/s2 * m = N * m = J
How about the elastic potential energy for a spring:
PEspring = 1/2 k *
2
x
now units on the right side:
N/m * m2 = N * m = J
Energy conservation
• This is one of the most important
principles we know in Physics - sort of the
“holy grail”
• It is also one of the most basic law’s fundamental
Definition
Conservation of Energy is the observation that for a system whose mass does
not change, energy is not created nor destroyed. Energy may change forms;
energy may be exchanged with the system's environment. Any change in the
total energy of the system is accounted for by the opposing change in energy
of the system's environment.
Within an isolated system, one type of energy can be transformed into another
type of energy, but the total of all energies in the system is constant. The energy
of a system changes by the work done on or by the system and the heat that
enters or leaves the system.
Etotal = (0.5)mv2 + mgh + (0.5)kx2 is unchanging if no work is done on or by the
system, and no heat is gained or lost..
Example
A 10.0 g pebble is placed in a sling shot with a spring constant of 200.0 N/m and is
stretched back 0.500 m. What is the maximum velocity the pebble will acquire? If shot
straight up, what is the maximum height the pebble will reach?
In each situation energy is conserved if heat effects and air resistance are negligible.
At the beginning, when the slingshot is "cocked"
Etotal= (0.5)mv2 + mgh + (0.5)kx2 = (0.5)m(0)2 + mg(0) + (0.5)(200.0 N/m)(0.500 m)2
= (0.25*100) N/m * m2 = 25 Nm
Etotal = 25 J
The maximum velocity is obtained after the pebble is released but before it has achieved
any altitude.
Etotal = (0.5)mv2 + mgh + (0.5)kx2
25 J = (0.5)(0.0100 kg)(v)2 + mg(0) + (0.5)k(0)2 --> v2 = 25J : 0.005 kg m = 5000 (m/s)2
v = 70.7 m/s
At its maximum height, the velocity of the pebble is zero.
Etotal = (0.5)mv2 + mgh + (0.5)kx2
25 J = (0.5)m(0)2 + (0.010 kg)(9.8 N/kg)h + (0.5)k(0)2
--> h = E : (m * g) = 25Nm : 0.098 N = 255 m
h = 255 m
Side note
What is a Perpetual Motion Machine?
A perpetual motion machine (PMM) is a device based on mechanical, chemical, electrical or other
physical processes which, when started, will remain in operation forever and provide additional
work as well. Only the natural wear of the components will eventually stop its operation. Consider
this definition, written in 1896!
In nature the motion of the sun and stars in the celestial sphere, and the recurring solstices
"proved" that eternal, or perpetual, motion is possible. These phenomena suggested the appealing
conclusion that man might reproduce God's creation in smaller scale. But this carried risk, for
perpetual motion seekers were regarded as heretics, and potential victims of the inquisition. On
the other hand, the idea of perpetual motion was an ambitious idea which promised to have great
value, and this provided some security. The Jesuits, the scientists of the church, worked diligently
on perpetual motion projects. This leads us directly to the heart of the history of a very special
kind of machine.
The idea of perpetual motion originated in the Orient. The first description of a perpetual motion
machine was found to be made by the Indian mathematician and astronomer Brahmagupta in 624. In
his Brahmasphutasiddhanta, he describes a perpetual motion device: "Make a wheel of light timber,
with uniformly hollow spokes at equal intervals. Fill each spoke up to half with mercury and seal its
opening situated in the rim. Set up the wheel so that its axle rests horizontally on two [upright]
supports. Then the mercury runs upwards [in some] hollow spaces and downwards [in some others, as
a result of which] the wheel rotates automatically forever." (transl. S. R. Sarma).
YOUR TURN:
1.) Jack and Jill, whose total mass is 120 kg, sit on a swing at the end of a 5m long
rope. Initially the rope attached to their swing makes an angle of 36 degrees with the
horizontal. At the bottom of the arc, Jill, whose mass is 52 kg, steps off. What is the
maximum height Jack can reach as the swing continues?
2.) How much heat is generated when the brakes are used to bring a 1000-kg car to rest
from a speed of 100 km/h?
Note: 1 kcal = 4186 J
The swinging couple start from a height of 5 - 5*cos(54°) = 2.06 m
Since energy is conserved, the maximum height Jack can return to
is 2.06 m.
Note: mgh = (1/2)mv2
so gh = (1/2)v2 ie. h is independent of mass.
How much kilocalories of heat are generated when the brakes are used to bring
a 1000-kg car to rest from a speed of 100 km/h?
(100 km/h)*(1000 m/km)*(1 h / 3600 s) = 27.8 m/s
Kinetic energy of the car = (1/2)(1000 kg)(27.8 m/s)2 = 3.86*105 J
This is the energy converted to heat as the car comes to a stop.
( 3.86*105 J)(1 kcal / 4.186*103 J) = 92.2 kcal
One more definition for today
Power
The quantity work has to do with a force causing a displacement. Work has
nothing to do with 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. The quantity which has to do
with 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.
Power is the rate at which work is done. It is the work/time ratio.
Mathematically, it is computed using the following equation.
P = W/t
The unit for power is Watt [W]
The standard metric unit of power is the Watt. As is implied by the equation
for power, a unit of power is equivalent to a unit of work divided by a unit of
time. Thus, a Watt is equivalent to a Joule/second. For historical reasons, the
horsepower is occasionally used to describe the power delivered by a machine.
One horsepower is equivalent to approximately 750 Watts.
James Watt, the son of a merchant, was born in Greenock, Scotland, in
1736. At the age of nineteen Watt was sent to Glasgow to learn the trade of
a mathematical-instrument maker.
After spending a year in London, Watt returned to Glasgow in 1757 where
he established his own instrument-making business. Watt soon developed
a reputation as a high quality engineer and was employed on the Forth &
Clyde Canal and the Caledonian Canal. He was also engaged in the
improvement of harbours and in the deepening of the Forth, Clyde and
other rivers in Scotland.
In 1763 Watt was sent a Newcomen steam engine to repair. While putting it
back into working order, Watt discovered how he could make the engine
more efficient. Watt worked on the idea for several months and eventually
produced a steam engine that cooled the used steam in a condenser
separate from the main cylinder.
A person is also a machine which has a power rating. Some people are more
power-full than others. That is, some people are capable of doing the same
amount of work in less time or more work in the same amount of time. A
common physics lab involves quickly climbing a flight of stairs and using mass,
height and time information to determine a student's personal power. Despite the
diagonal motion along the staircase, it is often assumed that the horizontal
motion is constant and all the force from the steps are used to elevate the
student upward at a constant speed. Thus, the weight of the student is equal to
the force which does the work on the student and the height of the staircase is
the upward displacement.
Suppose that Ben Pumpiniron elevates his 80-kg body up the 2.0 meter stairwell
in 1.8 seconds. If this were the case, then we could calculate Ben's power rating.
It can be assumed that Ben must apply a 800-Newton downward force upon the
stairs to elevate his body. By so doing, the stairs would push upward on Ben's
body with just enough force to lift his body up the stairs. It can also be assumed
that the angle between the force of the stairs on Ben and Ben's displacement is 0
degrees.
= 871 Watts
The expression for power is work/time. And since the expression for work is
force*displacement, the expression for power can be rewritten as
(force*displacement)/time. Since the expression for velocity is displacement/
time, the expression for power can be rewritten once more as force*velocity.
YOUR TURN:
1.) During a physics lab, Jack and Jill ran up a hill. Jack is twice as massive
as Jill; yet Jill ascends the same distance in half the time. Who did the most
work? Who delivered the most power? Explain your answers.
2.) Your household's monthly electric bill is often expressed in kilowatt-hours.
One kilowatt-hour is the amount of energy delivered by the flow of l kilowatt
of electricity for one hour. Use conversion factors to show how many joules of
energy you get when you buy 1 kilowatt-hour of electricity.
3.) An escalator is used to move 20 passengers every minute from the
first floor of a department store to the second. The second floor is located
5.20 meters above the first floor. The average passenger's mass is 54.9
kg. Determine the power requirement of the escalator in order to move
this number of passengers in this amount of time.
1.) Jack does more work than Jill. Jack must apply twice the force to lift
his twice-as-massive body up the same flight of stairs. Yet, Jill is just as
"power-full" as Jack. Jill does one-half the work yet does it one-half the
time. The reduction in work done is compensated for by the reduction in
time.
2.) Using conversion factors, it can be shown that 1 kilo-watt*hour is
equivalent to 3.6 x 106 Joules. First, convert 1 kW-hr to 1000 Watthours. Then convert 1000 Watt-hours to 3.6 x 106 Watt-seconds. Since a
Watt-second is equivalent to a Joule, you have found your answer.
3.) A good strategy would involve determining the work required to
elevate one average passenger. Then multiply this value by 20 to
determine the total work for elevating 20 passengers. Finally, the power
can be determined by dividing this total work value by the time required
to do the work. The solution goes as follows:
W1 passenger
W1 passenger = F • d • cos(0 deg)
= (54.9 kg • 9.8 m/s2) • 5.20 m = 2798 J (rounded)
W20 passengers = 55954 J (rounded)
P = W20 passengers / time = (55954 J) / (60 s)
P = 933 W