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AP Physics C - Mechanics
Simple Harmonic Motion
2015-12-05
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Table of Contents
Click on the topic to go to that section
· Spring and a Block
· Energy of SHM
· SHM and UCM
· Simple and Physical Pendulums
· Sinusoidal Nature of SHM
Spring and a Block
Return to Table
of Contents
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Periodic Motion
Periodic motion describes objects that oscillate about an
equilibrium point. This can be a slow oscillation - like the earth
orbiting the sun, returning to its starting place once a year. Or
very rapid oscillations such as alternating current or electric
and magnetic fields.
Simple harmonic motion is a periodic motion where there is
a force that acts to restore an object to its equilibrium point - it
acts opposite the force that moved the object away from
equilibrium.
The magnitude of this force is proportional to the displacement
of the object from the equilibrium point.
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Simple Harmonic Motion
Simple harmonic motion is described by Hooke's Law.
Robert Hooke was a brilliant scientist who helped survey and
architect London after the Great Fire of London in 1666, built
telescopes, vaccums, observed the planets, used microscopes
to study cells (the name cell comes from Hooke's observations
of plant cells) and proposed the inverse square law for
gravitational force and how this force explained the orbits of
planets.
Unfortunately for Robert Hooke, he was a contemporary of Sir
Isaac Newton and the two men were not friends. In fact, there
are no pictures of Hooke - possibly due to Newton's influence and Newton gave no credit to Hooke for any of his physics work.
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Hooke's Law
Hooke's Law
Hooke developed his law to explain the force that acts on an
elastic spring that is extended from its equilibrium (rest
position - where it is neither stretched nor compressed). If the
spring is stretched in the positive x direction, a restorative
force will act to bring it back to its equilibrium point - a negative
force:
k is the spring constant and its units are N/m.
For an object to be in simple harmonic motion, the force has to
be linearly dependent on the displacement. If it is proportional
to the square or any other power of the displacement, then the
object is not in simple harmonic motion.
The force is not constant, so the acceleration is not constant
either.
This means the kinematics equations cannot be used to solve
for the velocity or position of the object.
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1 A spring whose spring constant is 20N/m isstretched
1 A spring whose spring constant is 20N/m isstretched
0.20m from equilibrium; what is the magnitude of the
force exerted by the spring?
Answer
0.20m from equilibrium; what is the magnitude of the
force exerted by the spring?
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2 A spring whose spring constant is 150 N/mexerts a force
of 30N on the mass in a mass-spring system. How far is
the mass from equilibrium?
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2 A spring whose spring constant is 150 N/mexerts a force
of 30N on the mass in a mass-spring system. How far is
the mass from equilibrium?
Answer
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3 A spring exerts a force of 50N on the mass in a
3 A spring exerts a force of 50N on the mass in a
mass-spring system when it is 2.0m from
equilibrium. What is the spring's spring constant?
Answer
mass-spring system when it is 2.0m from
equilibrium. What is the spring's spring constant?
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Simple Harmonic Motion
The maximum force exerted on the mass is when the spring
is most stretched or compressed (x = -A or +A):
Simple Harmonic Motion
When the spring is all the way compressed:
F = -kA (when x = -A or +A)
x
The minimum force exerted on the mass is when the spring
is not stretched at all (x = 0)
-A
0
A
F = 0 (when x = 0)
· The displacement is at the negative amplitude.
· The force of the spring is in the positive direction.
x
-A
0
A
· The acceleration is in the positive direction.
· The velocity is zero.
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Simple Harmonic Motion
When the spring is at equilibrium and
heading in the positive direction:
Simple Harmonic Motion
When the spring is all the way stretched in the positive
direction:
x
-A
0
A
x
-A
0
A
· The displacement is zero.
· The displacement is at the positive amplitude.
· The force of the spring is zero.
· The force of the spring is in the negative direction.
· The acceleration is zero.
· The acceleration is in the negative direction.
· The velocity is positive and at a maximum.
· The velocity is zero.
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Simple Harmonic Motion
When the spring is at equilibrium and heading
in the negative direction:
x
-A
0
A
4 At which location(s) is the magnitude of theforce
on the mass in a mass-spring system a
maximum?
A
x=A
B
x=0
C x = -A
D x = A and x = -A
All of the above
E
· The displacement is zero.
· The force of the spring is zero.
· The acceleration is zero.
· The velocity is negative and at a maximum.
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4 At which location(s) is the magnitude of theforce
Answer
on the mass in a mass-spring system a
maximum?
A
x=A
B
x=0
C x = -A
D x = A and x = -A
All of the above
E
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5 At which location(s) is the magnitude of the force
on the mass in a mass-spring system a minimum?
A
B
C
D
E
x=A
x=0
x = -A
x = A and x = -A
All of the above
D
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5 At which location(s) is the magnitude of the force
A
B
C
D
E
x=A
x=0
x = -A
x = A and x = -A
All of the above
Answer
on the mass in a mass-spring system a minimum?
B
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Vertical Mass-Spring System
If the spring is hung
vertically, the only
change is in the
equilibrium position,
which is at the point
where the spring force
equals the gravitational
force.
y = y0
The displacement is now
measured from the new
equilibrium position, y = 0.
y=0
The value of k for an unknown spring can
be found via this arrangement.
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Vertical Mass-Spring System
Use Newton's Second Law
in the y direction when the
mass is at rest at its new
equilibrium position.
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6 An object of mass 0.45 kg is attached to a spring with
k = 100 N/m and is allowed to fall. What is the maximum
distance that the mass reaches before it stops and
begins heading back up?
ky0
mg
y = y0
y=0
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Springs in Parallel
6 An object of mass 0.45 kg is attached to a spring with
k = 100 N/m and is allowed to fall. What is the maximum
distance that the mass reaches before it stops and
begins heading back up?
Answer
Take a spring with spring constant k, and cut it in half.
What is the spring constant, k' of each of the two new
springs?
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Springs in Parallel
For a given applied force, mg, the new springs will stretch
only half as much as the original spring. Let y equal the
distance the springs stretch when the mass is attached.
Springs in Parallel
Next attach just one mass to the two spring combination. Let's
calculate the effective spring constant of two springs in parallel,
each with spring constant = k', by using a free body diagram.
y is the distance each spring is stretched.
ky
m
m
m
The spring constant of each piece is
twice the spring constant of the
original spring.
ky
m
mg
By cutting a spring in half, and then
attaching each piece to a mass, the effective
spring constant is quadrupled. The spring
system is four times as stiff as the original
spring.
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Springs in Parallel
Springs in Parallel
For identical springs in parallel, the effective spring constant is
just twice the spring constant of either spring. We cannot
generally apply this to springs with different spring constants..
If the springs had different spring constants, then one spring
would be stretched more than the other - and the mass would
feel a net torque and rotate. It would be hard to predict what
the behavior of the mass would be. So, the problems will be
limited to identical springs in parallel.
Why?
m
m
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Springs in Series
Springs in Series
We don't have this limitation for springs in series, as they
contact the mass at only one point. Take two springs of spring
constants k 1 and k 2, and attach them to each other. For a given
force, each spring stretches a distance y 1 and y 2 where the total
stretch of the two springs is y T.
y1
yT = y1 + y2
F is given and constant
y1
The effective spring constant of
the two springs in series is:
y2
yT = y1 + y2
m
y2
m
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keff is less than either one of the
spring constants that were joined
together. The combination is less
stiff then either spring alone with
the mass.
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Energy of SHM
The spring force is a conservative force which allows us to
calculate a potential energy associated with simple harmonic
motion.
The force is not constant, so in addition to not being able to use
the kinematics equations to predict motion, the potential energy
can't be found by taking the negative of the work done by the
spring on the block where work is found by multiplying a
constant force by the displacement.
Energy of SHM
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of Contents
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Energy of SHM
Elastic Potential Energy
Integral Calculus!
At each point of the spring's motion, the force is different. In
order to calculate work, the motion must be analyzed at
infinitesimal displacements which are multiplied by the force at
each infinitesimal point, and then summed up.
Start at the equilibrium point,
x0 = 0, and stretch the spring to xf.
What does that sound like?
EPE has been used in this
course, but U is generally the
symbol for potential energy.
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Energy in the Mass-Spring System
Energy and Simple Harmonic Motion
There are two types of energy in a mass-spring system.
Any vibrating system where the restoring force is
proportional to the negative of the displacement is in
simple harmonic motion (SHM), and is often called a
simple harmonic oscillator.
The energy stored in the spring because it is stretched or
compressed:
Also, SHM requires that a system has two forms of
energy and a method that allows the energy to go back
and forth between those forms.
AND
The kinetic energy of the mass:
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Energy in the Mass-Spring System
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EPE
At any moment, the total energy of the system is
constant and comprised of those two forms.
EPE
When the mass is at the limits of its
motion (x = A or x = -A), the energy is all
potential:
When the mass is at the equilibrium
point (x=0) the spring is not stretched
and all the energy is kinetic:
The total mechanical energy is constant.
EPE
But the total energy is constant.
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Energy in the Mass-Spring System
Energy in the Mass-Spring System
When the spring is passing through the equilibrium....
When the spring is all the way compressed....
E (J)
ET
KE
UE
E (J)
· EPE is at a
maximum.
· EPE is zero.
ET
KE
UE
· KE is zero.
· Total energy is
constant.
· Total energy is
constant.
x (m)
x (m)
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Energy in the Mass-Spring System
When the spring is all the way stretched....
KE
UE
7 At which location(s) is the kinetic energy of a
mass-spring system a maximum?
A
B
C
D
E
E (J)
ET
· KE is at a maximum.
x=A
x=0
x = -A
x = A and x = -A
All of the above
· EPE is at a
maximum.
· KE is zero.
· Total energy is
constant.
x (m)
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7 At which location(s) is the kinetic energy of a
mass-spring system a maximum?
8 At which location(s) is the spring potentialenergy
(EPE) of a mass-spring system a maximum?
A
B
C
D
E
x=A
x=0
x = -A
x = A and x = -A
All of the above
Answer
A
B
C
D
E
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B
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x=A
x=0
x = -A
x = A and x = -A
All of the above
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8 At which location(s) is the spring potentialenergy
(EPE) of a mass-spring system a maximum?
x=A
x=0
x = -A
x = A and x = -A
All of the above
9 At which location(s) is the total energy of a mass-
spring system a maximum?
A
B
C
D
E
Answer
A
B
C
D
E
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x=A
x=0
x = -A
x = A and x = -A
All of the above
D
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9 At which location(s) is the total energy of a mass-
spring system a maximum?
x=A
x=0
x = -A
x = A and x = -A
All of the above
10 At which location(s) is the kinetic energy of a mass-
spring system a minimum?
A
B
C
D
E
Answer
A
B
C
D
E
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x=A
x=0
x = -A
x = A and x = -A
All of the above
E
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10 At which location(s) is the kinetic energy of a mass-
A
B
C
D
E
x=A
x=0
x = -A
x = A and x = -A
All of the above
Answer
spring system a minimum?
D
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11 What is the total energy of a mass-spring system if
11 What is the total energy of a mass-spring system if
the mass is 2.0kg, the spring constant is 200N/m
and the amplitude of oscillation is 3.0m?
Answer
the mass is 2.0kg, the spring constant is 200N/m
and the amplitude of oscillation is 3.0m?
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12 What is the maximum velocity of the mass in the
12 What is the maximum velocity of the mass in the
mass-spring system from the previous slide: the
mass is 2.0kg, the spring constant is 200N/m and
the amplitude of oscillation is 3.0m?
Answer
mass-spring system from the previous slide: the
mass is 2.0kg, the spring constant is 200N/m and
the amplitude of oscillation is 3.0m?
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The Period and Frequency
of a Mass-Spring System
We can use the period and frequency of a particle moving
in a circle to find the period and frequency:
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13 What is the period of a mass-spring system if the
mass is 4.0kg and the spring constant is 64N/m?
Slide 48 (Answer) / 102
13 What is the period of a mass-spring system if the
14 What is the frequency of the mass-spring system
from the previous slide; the mass is 4.0kg and the
spring constant is 64N/m?
Answer
mass is 4.0kg and the spring constant is 64N/m?
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14 What is the frequency of the mass-spring system
from the previous slide; the mass is 4.0kg and the
spring constant is 64N/m?
Answer
SHM and UCM
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SHM and Circular Motion
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Period
There is a deep connection between Simple Harmonic
Motion (SHM) and Uniform Circular Motion (UCM).
The time it takes for an object to complete one trip around
a circular path is called its Period.
Simple Harmonic Motion can be thought of as a onedimensional projection of Uniform Circular Motion.
The symbol for Period is "T"
All the ideas we learned for UCM, can be applied to
SHM...we don't have to reinvent them.
So, let's review circular motion first, and then extend what
we know to SHM.
Click here to see how circular motion
relates to simple harmonic motion.
Periods are measured in units of time; we will usually use
seconds (s).
Often we are given the time (t) it takes for an object to make
a number of trips (n) around a circular path. In that case,
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15 If it takes 50 seconds for an object to travel around
15 If it takes 50 seconds for an object to travel around
a circle 5 times, what is the period of its motion?
Answer
a circle 5 times, what is the period of its motion?
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16 If an object is traveling in circular motion and its
16 If an object is traveling in circular motion and its
period is 7.0s, how long will it take it to make 8
complete revolutions?
Answer
period is 7.0s, how long will it take it to make 8
complete revolutions?
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Frequency
The number of revolutions that an object completes in a
given amount of time is called the frequency of its motion.
The symbol for frequency is "f"
Periods are measured in units of revolutions per unit time;
we will usually use 1/seconds (s -1 ). Another name for s -1 is
Hertz (Hz). Frequency can also be measured in
revolutions per minute (rpm), etc.
Often we are given the time (t) it takes for an object to
make
a number of revolutions (n). In that case,
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17 An object travels around a circle 50 times in ten
seconds, what is the frequency (in Hz) of its
motion?
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17 An object travels around a circle 50 times in ten
18 If an object is traveling in circular motion with a
frequency of 7.0 Hz, how many revolutions will it
make in 20s?
Answer
seconds, what is the frequency (in Hz) of its
motion?
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18 If an object is traveling in circular motion with a
Answer
frequency of 7.0 Hz, how many revolutions will it
make in 20s?
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Period and Frequency
Since
and
then
and
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19 An object has a period of 4.0s, what is the
frequency of its motion (in Hz)?
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20 An object is revolving with a frequency of 8.0 Hz,
what is its period (in seconds)?
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Velocity
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21 An object is in circular motion. The radius of its
motion is 2.0 m and its period is 5.0s. What is its
velocity?
Also, recall from Uniform Circular Motion....
and
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22 An object is in circular motion. The radius of its
motion is 2.0 m and its frequency is 8.0 Hz. What is
its velocity?
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22 An object is in circular motion. The radius of its
motion is 2.0 m and its frequency is 8.0 Hz. What is
its velocity?
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SHM and Circular Motion
In UCM, an object
completes one circle, or
cycle, in every T seconds.
That means it returns to
its starting position after T
seconds.
Answer
In Simple Harmonic
Motion, the object does
not go in a circle, but it
also returns to its starting
position in T seconds.
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Any motion that repeats over and over again, always
returning to the same position is called " periodic".
Click here to see how simple harmonic
motion relates to circular motion.
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23 It takes 4.0s for a system to complete one cycle of
simple harmonic motion. What is thefrequency of
the system?
24 The period of a mass-spring system is 4.0s and
the amplitude of its motion is 0.50m. How fardoes
the mass travel in 4.0s?
Slide 66 (Answer) / 102
24 The period of a mass-spring system is 4.0s and
the amplitude of its motion is 0.50m. How fardoes
the mass travel in 4.0s?
Answer
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25 The period of a mass-spring system is 4.0s and the
25 The period of a mass-spring system is 4.0s and the
amplitude of its motion is 0.50m. How fardoes the
mass travel in 6.0s?
Answer
amplitude of its motion is 0.50m. How fardoes the
mass travel in 6.0s?
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· Displacement is measured from the
equilibrium point
· Amplitude is the maximum
displacement (equivalent to the radius, r, in
UCM).
· A cycle is a full to-and-fro motion (the
same as one trip around the circle in
UCM)
· Period is the time required to complete
one cycle (the same as period in UCM)
Simple and Physical
Pendulums
· Frequency is the number of cycles
completed per second (the same as
frequency in UCM)
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The Simple Pendulum
A simple pendulum consists of a mass at the end of
a lightweight cord. We assume that the cord does
not stretch, and that its mass is negligible.
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The Simple Pendulum
In order to be in SHM, the restoring force
must be proportional to the negative of
the displacement. Here we have:
which is proportional to sin θ and not to θ
itself.
We don't really need to worry about this
because for small angles (less than 15
degrees or so), sin θ ≈ θ and x = Lθ. So
we can replace sin θ with x/L.
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The Simple Pendulum
has the form of
if
26 What is the frequency of the pendulum of the
previous slide (a length of 2.0m near the surface of
the earth)?
But we learned before that
Substituting for k
Notice the "m" canceled out, the mass doesn't matter.
Slide 73 (Answer) / 102
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The Simple Pendulum
26 What is the frequency of the pendulum of the
previous slide (a length of 2.0m near the surface of
the earth)?
Answer
So, as long as the cord can
be considered massless
and the amplitude is small,
the period does not depend
on the mass.
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a pendulum?
A
B
C
D
E
the acceleration due to gravity
the length of the string
the mass of the pendulum bob
A&B
A&C
27 Which of the following factors affect the period of
a pendulum?
A
B
C
D
E
the acceleration due to gravity
the length of the string
the mass of the pendulum bob
A&B
A&C
Answer
27 Which of the following factors affect the period of
Slide 75 (Answer) / 102
D
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Energy in the Pendulum
The two types of energy in a pendulum are:
Gravitational Potential Energy
Slide 77 / 102
Energy in the Pendulum
At any moment in time the total energy of the system is
contant and comprised of those two forms.
AND
The kinetic energy of the mass:
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28 What is the total energy of a 1 kg pendulum if
Slide 78 (Answer) / 102
28 What is the total energy of a 1 kg pendulum if
its height, at its maximum amplitude is 0.20m
above its height at equilibrium?
Answer
its height, at its maximum amplitude is 0.20m
above its height at equilibrium?
The total mechanical energy is constant.
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29 What is the maximum velocity of the
pendulum's mass from the previous slide (its
height at maximum amplitude is 0.20m above
its height at equilibrium)?
Slide 79 (Answer) / 102
29 What is the maximum velocity of the
pendulum's mass from the previous slide (its
height at maximum amplitude is 0.20m above
its height at equilibrium)?
Answer
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Position as a function of time
The position as a function of for an object in simple harmonic
motion can be derived from the equation:
Sinusoidal Nature of
SHM
Where A is the amplitude of oscillations.
Take note that it doesn't really matter if you are using sine or
cosine since that only depends on when you start your clock. For
our purposes lets assume that you are looking at the motion of a
mass-spring system and that you start the clock when the mass is
at the positive amplitude.
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of Contents
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Position as a function of time
Now we can derive the equation for position as a function of time.
Since
we can replace θ with ωt.
And we can also replace ω with 2πf or 2π/T.
Where A is amplitude, T is period, and t is time.
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Velocity as a function of time
We can also derive the equation for velocity as a function of time.
Since v=ωr can replace v with ωA as well as θ with ωt.
And again we can also replace ω with 2πf or 2π/T.
Where A is amplitude, T is period, and t is time.
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Slide 86 / 102
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Acceleration as a function of time
We can also derive the equation for acceleration as a function of
time.
Since a=rω2 can replace a with A ω2 as well as θ with ωt.
And again we can also replace ω with 2πf or 2π/T.
Where A is amplitude, T is period, and t is time.
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The Sinusoidal Nature of SHM
Now you can see all of the
graphs together.
Take note that when the
position is at the positive
amplitude, the acceleration
is negative and the velocity
is zero.
Or when the velocity is at a
maximum both the position
and acceleration are zero.
http://www.youtube.com/watch?
v=eeYRkW8V7Vg&feature=Play
List&p=3AB590B4A4D71006
&index=0
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The Period and Sinusoidal Nature of SHM
The Period and Sinusoidal Nature of SHM
Use this graph to answer the following questions.
a
(acceleration)
a
(acceleration)
v (velocity)
x (displacement)
v (velocity)
x (displacement)
T/4
T/2
3T/4
T
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30 What is the acceleration when x = 0?
31 What is the acceleration when x = A?
A
a<0
a (acceleration)
A
a<0
a (acceleration)
B
a=0
B
a=0
C
a>0
v (velocity)
C
a>0
v (velocity)
D
It varies.
x (displacement)
D
It varies.
x (displacement)
T/4
T/2
3T/4
T/4
T
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a<0
B
a=0
C
D
3T/4
T
Slide 95 / 102
32 What is the acceleration when x = -A?
A
T/2
33 What is the velocity when x = 0?
a (acceleration)
A
v<0
a (acceleration)
B
v=0
a>0
v (velocity)
C
v>0
v (velocity)
It varies.
x (displacement)
D
A or C
x (displacement)
T/4
T/2
3T/4
T/4
T
Slide 96 / 102
v<0
B
v=0
C
D
3T/4
T
Slide 97 / 102
34 What is the velocity when x = A?
A
T/2
35 Where is the mass when acceleration is at amaximum?
a (acceleration)
A
x=A
a (acceleration)
B
x=0
v>0
v (velocity)
C
x = -A
v (velocity)
A or C
x (displacement)
D
A or C
x (displacement)
T/4
T/2
3T/4
T
T/4
T/2
3T/4
T
Slide 98 / 102
Slide 99 / 102
36 Where is the mass when velocity is at a maximum?
A
x=A
B
x=0
C
x = -A
v (velocity)
D
A or C
x (displacement)
37 Which of the following represents the position as a
function of time?
a (acceleration)
a (acceleration)
v (velocity)
x (displacement)
T/4
T/2
3T/4
T
T/4
T/2
3T/4
A
x = 4 cos (2t)
D
x = 8 cos (2t)
B
x = 2 cos (2t)
C
x = 2 sin (2t)
Slide 100 / 102
38 Which of the following represents the velocity as a
function of time?
T/4
T/2
3T/4
T
Slide 101 / 102
39 Which of the following represents the acceleration
as a function of time?
a (acceleration)
a (acceleration)
v (velocity)
v (velocity)
x (displacement)
x (displacement)
T/4
T
T/2
3T/4
T
A
v = -12 sin (2t)
D
v = -4 sin (2t)
A
v = -8 sin (2t)
D
v = -4 sin (2t)
B
v = -12 cos (2t)
C
v = -4 cos (2t)
B
v = -8 cos (2t)
C
v = -4 cos (2t)
Slide 102 / 102