Download v - Purdue Physics

5th Week Lectures
Feb. 06.- Feb. 10. 2017.
Work, Energy, Power
Energy makes our everyday world work
Where does energy go?
Are we using it up?
How can one store energy?
Where does energy come from.
The heat of the earth
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Physics 214 Spring 2017
1
Announcements
Exam 1
Feb. 23. 2017. Thursday
8:00-10:00pm
in Phys. Room 112
Exam Calculator: When taking a Physics 214 Exam, there is only one
calculator model that is acceptable: The CASIO fx-260 SLRSC FRACTION.
NO OTHER BRAND or TYPE WILL BE ALLOWED!
Please bring your I.D. card and #2 pencil
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Physics 214 Spring 2017
2
Work, Energy and Power
We all use the words Work, Energy and Power in our everyday
language .
However, we need to write down simple mathematical definitions
so that we will be able to do calculations.
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Energy Conservation
If we take a closed system, that is one that no energy can enter or
leave, then there is a physical law which states that energy of a
closed system is conserved.
We will define various forms of energy and we examine the
system as a function of time. Energy may change into different
forms but the total is constant. Energy does not have direction
just a magnitude and units.
Thus energy is a scalar quantity.
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Forms of mechanical energy
One obvious form of energy is the energy
of a moving object.
2
mv
This is called Kinetic Energy Ek 
2
A second form of energy is what is called Potential Energy E p .
This energy is the energy stored in a compressed spring or
stretched bowstring or in an object that is held at rest above the
earth’s surface.
When the spring or bowstring released one gets kinetic energy
appearing from the stored energy. In the case of a pendulum there
is a continual storage of potential energy and its subsequent
release into kinetic energy as the pendulum swings.
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Work and energy
If an object is initially at rest
and a net force F is acted on it,
it will accelerate and after time t, the object
will have moved a distance d .
+
F
d
We define Work as W  Fd , in units of joules, where F  d  F  d  cos  .
F  d is called a scalar product of two vectors.
Since cos  can vary between 1  cos   1
W can be positive or negative.
Now take our usual equations with the
assumption v0  0.

v = v0 + at d = v0t +1/2at2 and F = ma
Fd = ma(1/2at2) = ma(1/2av2/a2) = 1/2mv2 kinetic energy
F is the net force in the direction of motion
F = 50Ncos = 40N
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x
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Negative Work
If F is in the opposite direction to
the motion then cos   cos180  1
thus Fd is negative.
F
d
Remember: F and d are vectors
thus have magnitude and direction
and can be parallel or antiparallel.
F
Ff
If the work is negative, energy is being
removed from the moving object .
Friction always opposes motion and the
work Ff does is negative
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W = Fd - Ffd
Physics 214 Spring 2017
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Net force and Work
If there is more than one
force acting we have to
find the work done by each
force and the work done by
the net force
Net force
F – Ff
F
Ff
d
work = (F – Ff)d = 1/2mv2
The work the force F does is Fd and if we write the equation as
Fd = Ffd + 1/2mv2
we can see that some work goes into heat and some into kinetic
energy and we can account for all the work and energy
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1M-04 Pile Driver
The kinetic energy of a pendulum is transferred to a block which then slides to rest
What happens to
the Potential
Energy of the
Mass M ?
The potential energy of the pendulum
is turned into kinetic energy. Then if
the collision is perfectly elastic all the
kinetic energy is transferred to the
block and then the energy is turned
into heat through friction.
Mgh = Ffd
Ff is the average frictional force
between the block and the wood.
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1M-05 Pile Driver
Pile driver is used to pound structural support beam into the ground.
We illustrate this idea on a nail.
A Pile Driver does work on a nail
What happens to
the Potential
Energy of the
Mass M ?
Work-Energy Relationship
mg(h+y)
=
fy
Decrease of pot. energy = work done
f . is the average friction force
between the nail and the wood.
POTENTIAL ENERGY CHANGES TO KINETIC ENERGY,
KINETIC ENERGY CHANGES TO WORK.
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Potential Energy ( E p )
If we raise an object a height h so that it
starts and finishes at rest then the average
force = mg and the work done = mgh.
This energy is stored as potential energy
since if the mass is allowed to fall back to
it’s original point then
h
F = mg
g
v2 = v02 + 2gh
If v02  0 , after multiplying both sides by m
2
we have
mgh = 1/2mv2
So the original work in lifting is
stored and then returned as
kinetic energy
Similarly for a spring stored energy E=p  mgh
1/2kx2
Where x is the distance stretched
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Physics 214 Spring 2017
Ep 
1 2
kx
2
11
Potential energy
mv 2
EK 
 0 if
2
(Ep )
v0
d
Unlike kinetic energy for Potential energy we have to
define where zero is.
h
A block is at a height h above the floor and d above
the desk.
Potential energy is mgh with respect to the floor but
mgd with respect to the desk. If we dropped the block
it would have more kinetic energy hitting the floor
than hitting the desk.
Pole - Vaulting
An other examples is pole-vaulting, where kinetic
energy of running is converted to potential energy by
bending the pole and finally the potential energy of the
bent pole lifts the jumper over the bar.
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Oscillations
Many simple systems oscillate with a
continual transfer from
Ek to E p and E p to Ek
with the sum of the two energies remaining
constant.
In practice energy is lost through friction
and the motion slows down.
http://www.physics.purdue.edu/class/
applets/phe/pendulum.htm
http://www.physics.purdue.edu/academic_programs/courses/phys214/movies.php (anim0006.mov) (anim0007.mov) (anim0009.mov)
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1M-01 Bowling Ball Pendulum
A bowling ball attached to a wire is released like a pendulum
Is it safe to
stand here after
I release the
bowling ball ?
mgh
mgh
h
1/2mv2
mgh = 1/2 mv2
NO POSITIVE WORK IS DONE ON THE BALL
THUS, THERE IS NO GAIN IN TOTAL ENERGY
THE BALL WILL NOT GO HIGHER THAN THE INITIAL POSITION
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1M-03 Triple Chute
Three Steel Balls travel down different Paths
Each path is
clearly different.
Which ball will
travel the
farthest ?
The Change in Gravitational
Potential Energy does not
depend on the Path Traveled
EACH BALL HAS SAME KINETIC ENERGY AT BOTTOM OF
RAMP, REGARDLESS OF THE PATH TAKEN AND HAS THE
SAME VELOCITY
EACH OF THE STEEL BALLS LANDS AT THE SAME
POSITION
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Conservative forces
Gravity is an example of a conservative
force where total energy is conserved
and there is just an interchange
between kinetic and potential energy.
In real life frictional forces would cause
energy to be lost as heat
For a conservative force if no energy is
added or taken out then
Etotal  E p  Ek  constant = total energy is conserved
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1M-08 Galileo Track
Ball travels down one ramp and up a much steeper ramp
Will the ball travel
to a lower or higher
height when going
up the steeper,
shorter ramp ?
Conservation of Energy:
mgh = 1/2mv2 = mgh
So, The Ball should return
to the same height
AS THE BALL OSCILLATES BACK AND FORTH, THE
HEIGHT IS REDUCED BY A LITTLE. WHAT MIGHT
ACCOUNT FOR THIS?
FRICTION IS SMALL, BUT NOT ZERO.
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1M-10 Loop-the-Loop
Ball travels through a Loop-the-Loop
From what height
should the ball be
dropped to just
clear the Loop-theLoop ?
top
Conservation of Energy:
mgH = mg(2R) + 1/2mv2
(Eq. 1)
At the top of the loop
N + mg = mv2/R
(Eq.2)
The minimum speed is when N = 0
mv 2
Thus the top the minimal velocity is mg 
R
Substituting for
mv 2
2

mv 2 Rmg

2
2
from Eq.3 into Eq.1 we obtain mgH  mg (2 R) 
(Eq. 3)
Rmg
2
Therefore H = 5/2R (Friction means in practice H must be a litter larger)
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Power
It is not only important how much
work is done but also the rate at
which work is done
So the quantity
Power = P = W/t (unit is a watt)
is very important.
Generally energy supplies, motors
etc are rated by power and one
can determine how much work
can be done by multiplying by time.
W = Pt
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(joules)
Physics 214 Spring 2017
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Watts and Joules
Joule is the Unit of Energy and Energy is the fundamental
resource that is required for all activity and for life itself. All our
energy comes from the sun although there is geothermal energy
which was produced by the formation of the earth and tidal motion
produced by the motion of the moon.
Practical The unit for electrical usage is the kilowatt –hour. A
kilowatt – hour is the energy used by a 1000 watt device for 3600
seconds 1kWHr = 1000*3600 = 3.6 million joules
Watt is the Unit of Power and Power measures the rate at which
work is done or energy is used. All appliances, motors etc are rated
in Watts so that one can match to the required application.
1 horsepower = 1hp= 746Watts
Example. In order to lift an elevator with a mass of 1000kg to 100
meters requires 1000*9.8*100 joules but we need to do it in say 20
seconds so the power we need is 1000*9.8*100/20 = 49000 Watts so
we need to install a motor rated at > 49000 watts
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Mechanical Advantage
Very often we are limited by the maximum
force we can apply and the power we can
supply. This is also true of electric motors.
One can design simple arrangements so
that for example one can lift a large weight
by using a lever or a pulley system that
reduces the force.
The total work done is the same as lifting
the weight directly but for example using a
force which is half the weight but pulling it
for twice the distance
http://www.physics.purdue.edu/class/applets/phe/
pulleysystem.htm
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Equinox and Solstice
The Earth’s axis of rotation is not perpendicular the plane of Earth’s orbit around the Sun.
Instead the Earth’s rotational axis is tilted by 23.5  away from the perpendicular.
The Earth constantly maintains this tilted orientation as it orbits the Sun.
Seasonal changes, we experience on the Earth during a year, result from this tilt.
There are two days when both lengths of the day and night are equal ( Equinoxes = equal
nights):
The vernal (Spring) equinox on March 21. This marks the beginning of Spring in the
northern hemisphere.
The Autumn equinox on September 22. This marks the beginning of the Autumn on the
northern hemisphere.
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Between the vernal and autumnal equinoxes of the Earth there are two other significant
locations of the Sun: Summer solstice (summer solar standstill ) on June 21,
when summer begins in the northern hemisphere. It is the longest daylight day.
The Winter solstice (winter standstill ) on December 21 .
It is the shortest daylight day around Christmas.
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Optional Reading
Equinox Occurs on Sep. 22rd and March 21st when the sun is directly
overhead at the equator and day and night are of equal length
http://www.rkm.com.au/ANIMATIONS/animation-seasons.html
http://www.classzone.com/books/earth_science/terc/content/visualizations/es0408/es0408page01.cfm?chapter_no=04
http://www.mesoscale.iastate.edu/agron206/animations/01_EarthSun.html
The Tropic of Capricorn, or Southern tropic, marks the most southerly
latitude at which the sun can appear directly overhead at noon. This event
occurs at the December solstice, when the Southern hemisphere is tilted
towards the sun to its maximum extent. On the Northern hemisphere it is
the shortest daylight at X-mas.
The Tropic of Cancer, also referred to as the Northern tropic, is the circle
of latitude on the Earth that marks the most northerly position at which the
Sun may appear directly overhead at its zenith. This event occurs once
per year, at the time of the June solstice, when the Northern Hemisphere is
tilted toward the Sun to its maximum extent and has the longest daylight.
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Where do we get energy?
 Power comes from the sun 1.35 kilowatts/m2 on the atmosphere
and a maximum of about 1 kilowatt/m2 on earth. In one hour 1
kilowatt = 3600 x 103 joules. A toaster is usually 1 to 2 kilowatts.
Burning fossil fuels and making new molecules
carbon plus oxygen gives CO2 plus energy
 Nuclear power plants breaking heavy nuclei into lighter nuclei
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The heat of the earth
First we have to define what heat is.
Heat is the internal energy stored in an object by the motion
of it’s constituent particles (e.g. atoms)
How do we get heat in our everyday life?
We can transfer mechanical energy of an object into heat.
For example if drop a brick the kinetic energy just before
impact is turned into heat.
An object can also be heated by bombarding it with
particles of which photons from the sun is a common example.
That is why snow and ice can melt even if the temperature is below freezing
About 60% of the heat in the earth comes from the original
formation due to loss of potential energy and impact of the
material that makes up the earth.
About 40% comes from energy emitted in radioactive decays
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Summary of Chapter 6
W = Fd joules and can be + or –
F
Power = W/t
KE =
1/2mv2
watts
F
joules
PE = mgh or 1/2 kx2 joules
d
d
Conservative E = KE + PE
Gravity, oscillations such as a pendulum
or mass on a spring and KE and PE just
keep interchanging
http://www.physics.purdue.edu/class/applets/p
he/springpendulum.htm
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Questions Chapter 6
Q1 Equal forces are used to move blocks A and B across the
floor. Block A has twice the mass of block B, but block B moves
twice the distance moved by block A. Which block, if either, has
the greater amount of work done on it? Explain.
Work is Force times distance so the most work is done on B
Q3 A string is used to pull a wooden block
across the floor without accelerating the block.
The string makes an angle to the horizontal.
A. Does the force applied via the string do work
on the block?
F
d
B. Is the total force involved in doing work or just
a portion of the force?
A. Yes B. just the horizontal component
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Q4 In the situation pictured in question 3, if there is a frictional
force opposing the motion of the block, does this frictional
force do work on the block? Explain.
Yes it does negative work since force is opposite the motion
Q8 A woman uses a pulley, arrangement to lift a heavy crate. She
applies a force that is one-fourth the weight of the crate, but moves
the rope a distance four times the height that the crate is lifted. Is
the work done by the woman greater than, equal to, or less than the
work done by the rope on the crate? Explain.
The product Fd is the same for both and the work is equal
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Q12 A child pulls a block across the floor with force applied by a
horizontally held string. A smaller frictional force also acts upon
the block, yielding a net force on the block that is smaller than the
force applied by the string. Does the work done by the force
applied by the string equal the change in kinetic energy in this
situation?
No energy because is lost to friction. Fd – Ffd = 1/2mv2
Q18 Suppose that work is done on a large
crate to tilt the crate so that it is balanced on
one edge, as shown in the diagram, rather than
sitting squarely on the floor as it was at first.
Has the potential energy of the crate increased
in this process?
Yes. Work has been put in and the center of mass is now higher
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Q22 A pendulum is pulled back from its equilibrium (center)
position and then released.
A. What form of energy is added to the system prior to its release?
B. At what points in the motion of the pendulum after release is its
kinetic energy the greatest?
C. At what point is the potential energy the greatest?
A. Potential
B. at it’s lowest point
C. At the highest points where it stops
Q28 Suppose that a mass is hanging vertically at the end of a
spring. The mass is pulled downward and released to set it into
oscillation. Is the potential energy of the system increased or
decreased when the mass is lowered?
The potential energy is increased
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Ch 6 E 2
Woman does 160 J of work to move table 4m horizontally.
What is the magnitude of horizontal force applied?
F
d
Force & displacement in SAME direction
W = Fd,
160J = F(4m)
F = 40N
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Ch 6 E 8
5.0 kg box lifted (without acceleration) thru height of 2.0 m
a) What is increase in potential energy?
b) How much work was required to lift box?
a) PE = mgh PE = PEfinal – PEinitial
= mg(ho+2.0m) – mgho = mg(2.0m)
= (5.0 kg)(9.8 m/s2)(2.0m) = 98J
b) F = ma = 0 = Flift – mg
Flift = mg = (5.0kg)(9.8m/s2) = 49N
W = Fd = (49N)(2.0m) = 98J
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Physics 214 Spring 2017
M
ho+2.0m
M
g
Flift
M
mg
34
Ch 6 E 10
To stretch a spring a distance of 0.70 m, 40 J of work
is done.
What is the increase in potential energy?
b) What is the value of the spring constant k?
x=0
x=0.70 m

a) PE = 40J
equilibrium
b) PE = ½ kx2
k = 2PE/x2 = 80/(0.7)2 - = 296.8n/m
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Ch 6 E 18
The frequency of oscillation of a pendulum is 8 cycles/s.
What is its period?
x
T
f = 1/T
T = 1/f = 1/(8 cycles/s)
T = 0.125 seconds
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Physics 214 Spring 2017
t
36
Ch 6 CP 2
100 kg crate accelerated by net force = 50 N applied for 4 s.
a) Use Newton’s 2nd Law to find acceleration?
b) If it starts from rest, how far does it travel in 4 s?
c) How much work is done if the net force = 50 N?
a) F = ma a = F/m = 50N/100kg = 0/5 m/s2
M
Fnet
b) d = v0t + ½at2 = ½(0.5)(4)2 = 4m
c) W = Fd = (50N)(4m) = 200J
d) v = v0 + at = 0 + (0.5 m/s2)(4s) = 2m/s
e) KE = ½mv2 = ½(100kg)(2m/s)2 = 200 J
work done equals the kinetic energy.
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Ch 6 CP 4
A 0.20 kg mass is oscillating horizontally on a
friction-free table on a spring with a constant of
k=240 N/m. The spring is originally stretched to 0.12
m from equilibrium and released.
a) What is its initial potential energy?
b) What is the maximum velocity of the mass?
Where does it reach this maximum velocity?
c) What are values of PE, KE and velocity of mass
when the mass is 0.06 m from equilibrium.
d) What is the ratio of velocity in (c) to velocity in (b)
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Ch 6 CP 4 (con‘t)
a) PE = 1/2kx2 = ½(240)(0.12)2 = 1.73J
x=0
x=0.12 m
b) No friction so energy is conserved 
E=PE+KE, maximum KE when PE=0
KEmax = 1/2mv2
v = 4.16 m/s.
M
This occurs at the equilibrium position
c) PE = 1/2kx2 = ½(240)(0.06)2 = 0.432J
Since total energy = 1.73J then
the kinetic energy = 1.73 – 0.432 = 1.3J
KE = 1/2mv2 = 1.3 then v = 3.6m/s
d) vc/vb = 3.6/4.16 = 0.86
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Review Chapters 1 - 6
-
d
+ x
Units----Length, mass, time SI units m, kg, second
Coordinate systems
Average speed = distance/time = d/t
Instantaneous speed = d/Δt
Vector quantities---magnitude and direction
Magnitude is always positive
Velocity----magnitude is speed
Acceleration = change in velocity/time =Δv/Δt
Force = ma Newtons
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Conversions, prefixes and
scientific notation
giga
1,000,000,000
109
billion
1 in
2.54cm
mega
1,000,000
106
million
1cm
0.394in
kilo
1,000
103
thousand
1ft
30.5cm
centi
1/100
10-
hundredth
1m
39.4in
thousandth
1km
0.621mi
1mi
5280ft
1.609km
1lb
0.4536kg
g =9.8
1kg
2.205lbs
g=9.8
0.01
3.281ft
2
milli
micro
1/1000
1/1,000,000
0.00
1
1/106
103
10-
millionth
6
nano
1/1,000,000,000
1/109
109
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billionth
Physics 214 Spring 2017
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Speed, velocity and acceleration
v = Δd/Δt
a = Δv/Δt
The magnitude of a is not related
to the magnitude of v
2 3
4
1
the direction of a is not related to
the direction of v
v = v0 + at constant acceleration
d = v0t + 1/2at2
d = 1/2(v + v0) t d,v0 v,a can be + or –
independently
v2 = v02 + 2ad
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One dimensional motion and gravity
v = v0 + at d = v0t + 1/2at2
v2 = v02 + 2ad
d = ½(v + v0)t
+
g = -9.8m/s2
+
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At the top v = 0 and t = v0/9.8
At the bottom t = 2v0/9.8
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Equations
v = v0 + at
d = v0t + 1/2at2
d = ½(v + v0)t
v2 = v02 + 2ad
Sometimes you have to use two equations.
`
h
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v0 = 15m/s v = 50m/s What is h?
v = v0 + at
v0
50 = 15 + 9.8t t = 3.57 s
h = v0t + 1/2at2
g
h = 15 x 3.57 + 1/2x9.8x3.572
= 116m
v
h = ½(15 + 50) x 3.57 = 116m
Physics 214 Spring 2017
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Projectile Motion
axis 1
axis 2
v1 = constant and d1 = v1t
vv = v0v + at
and d = v0vt + 1/2at2
v1
g
9.8m/s2
h
v
R
h = v0vt + 1/2at2
Use + down so g is + and h is +
v0v = 0,
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t2 = 2h/a
R = v1t
v = v0v + at
Physics 214 Spring 2017
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Complete Projectile
v0v
v1
9.8m/s2
v1
v1
v0v
highest point the vertical velocity is zero
vv = v0v + at
so t = v0v/9.8
h = v0vt + 1/2at2
end t = 2v0v/9.8 and R = v1 x 2v0v/9.8
and the vertical velocity is minus v0v
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Newton’s Second and First Law
Second Law F = ma unit is a Newton (or pound)
First Law
F = 0 a = 0 so v = constant
Third law For every force there is an equal and opposite
reaction force
N
Weight = mg
mg
Ff
F
F
Ff
F = ma
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v = v0 + at
d = v0t + ½ at2 d = ½(v + v0)t v2 = v02 + 2ad
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Examples
+
T
N
g
30 – 8 – T = 4a
T – 6 = 2a
30 – 8 – 6 = 6a
mg
N – mg = ma
a + N > mg
a – N < mg
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Forces
Forces are responsible for all physical phenomena
Gravitation and the electromagnetic force are responsible for all the
phenomena we normally observe in our everyday life.
Newton’s laws
v = v0 + at
F = ma where F is net force
d = v0t + ½ at2
d = ½(v + v0)t v2 = v02 + 2ad
Every force produces an equal and opposite reaction
Weight = mg where g = 9.8m/s2 locally
Apparent weight in an elevator depends on the acceleration
a up weight is higher
a down weight is lower
If your weight becomes zero it’s time to worry because you are in free fall!!
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Circular motion, gravitation
Ferris wheel
N
Ff
F = ma = mv2/r
v
Rear
Ff = mv2/r
W = mg
Bottom N - mg = mv2/r
top
Mg – N = mv2/r
Mg –N = mv2/r
Gravitation
GmM/r2 = mv2/r
v2 = GM/r
T = 2πr/v
T2 = 4π2r2/v2 = 4π2r3/GMs
T2/r3 = 4π2/GMs
2/12/2017
Physics 214 Spring 2017
50
Examples of circular motion
Vertical motion
Looking down
N
N
v
v
W = mg
mg – N =
N = mv2/r
mv2/r
Side
N
mg
N - mg = mv2/r
2/12/2017
v
T
mg
Ff
mg
mg = Ff
Physics 214 Spring 2017
mg + T = mv2/r top
T - mg = mv2/r bottom
51
Work energy and Power
Kinetic energy = 1/2mv2
W = Fd and can be + or –
F is net force parallel to d.
Units are joules
Power = W/t watts
F
v
d
Potential energy = mgh
Spring = 1/2kx2
h
Oscillations
Transfer of KE
F = mg
g
PE
Conservative force
Transfer of KE
PE
2/12/2017
Physics 214 Spring 2017
52