AP Physics C - Mechanicscontent.njctl.org/courses/science/ap-physics-c-mechanics/... · 2015. 12. 10. · Slide 8 / 102 Hooke's Law For an object to be in simple harmonic motion,

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AP Physics C - Mechanics

Simple Harmonic Motion

2015-12-05

www.njctl.org

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Table of Contents

· Energy of SHM

Click on the topic to go to that section

· Spring and a Block

· SHM and UCM· Simple and Physical Pendulums

· Sinusoidal Nature of SHM

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Spring and a Block

Return to Tableof Contents

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Periodic MotionPeriodic 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 MotionSimple 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 LawHooke 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.

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Hooke's Law

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 is stretched 0.20m from equilibrium; what is the magnitude of the force exerted by the spring?

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1 A spring whose spring constant is 20N/m is stretched 0.20m from equilibrium; what is the magnitude of the force exerted by the spring?

[This object is a pull tab]

Ans

wer

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2 A spring whose spring constant is 150 N/m exerts a force of 30N on the mass in a mass-spring system. How far is the mass from equilibrium?


2 A spring whose spring constant is 150 N/m exerts a force of 30N on the mass in a mass-spring system. How far is the mass from equilibrium?


Ans

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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?


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?


Ans

wer

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The maximum force exerted on the mass is when the spring is most stretched or compressed (x = -A or +A):

F = -kA (when x = -A or +A)

The minimum force exerted on the mass is when the spring is not stretched at all (x = 0)

F = 0 (when x = 0)


xA0-A

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· The displacement is at the negative amplitude.

· The force of the spring is in the positive direction.

· The acceleration is in the positive direction.

· The velocity is zero.

When the spring is all the way compressed:

xA0-A

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· The displacement is zero.

· The force of the spring is zero.

· The acceleration is zero.

· The velocity is positive and at a maximum.

When the spring is at equilibrium and heading in the positive direction:

xA0-A

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· The displacement is at the positive amplitude.

· The force of the spring is in the negative direction.

· The acceleration is in the negative direction.

· The velocity is zero.

When the spring is all the way stretched in the positive direction:

xA0-A

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· The displacement is zero.

· The force of the spring is zero.

· The acceleration is zero.

· The velocity is negative and at a maximum.

When the spring is at equilibrium and heading in the negative direction:

xA0-A

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4 At which location(s) is the magnitude of the force on the mass in a mass-spring system a maximum?

A x = AB x = 0C x = -AD x = A and x = -AE All of the above


4 At which location(s) is the magnitude of the force on the mass in a mass-spring system a maximum?



Ans

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D

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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?



5 At which location(s) is the magnitude of the force on the mass in a mass-spring system a minimum?



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B

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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.

The displacement is now measured from the new equilibrium position, y = 0.

Vertical Mass-Spring System

y = 0

y = y0

The value of k for an unknown spring can be found via this arrangement.

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Vertical Mass-Spring System

y = 0

y = y0

mg

ky0

Use Newton's Second Lawin 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?


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?


Ans

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Springs in Parallel

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

m m

m

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.

The spring constant of each piece is twice the spring constant of the original spring.

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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.

Springs in Parallel

m

mg

kyky

y is the distance each spring is stretched.

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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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..

Springs in Parallel

m

Why?

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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.

Springs in Parallel

m

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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 k2, and attach them to each other. For a given force, each spring stretches a distance y 1 and y2 where the total stretch of the two springs is y T.

Springs in Series

y1

y2

yT = y1 + y2

F is given and constant

m

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The effective spring constant of the two springs in series is:

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.

Springs in Series

y1

y2

yT = y1 + y2

m

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Energy of SHM


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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.

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Energy of SHM

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.

What does that sound like?

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Elastic Potential Energy

Start at the equilibrium point, x0 = 0, and stretch the spring to xf.

EPE has been used in this course, but U is generally the symbol for potential energy.

Integral Calculus!

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Energy and Simple Harmonic Motion

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.

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.

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Energy in the Mass-Spring System

There are two types of energy in a mass-spring system.

The energy stored in the spring because it is stretched or compressed:

AND

The kinetic energy of the mass:

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The total mechanical energy is constant.

At any moment, the total energy of the system is constant and comprised of those two forms.


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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:

But the total energy is constant.EPE

EPE

EPE

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When the spring is all the way compressed....


· EPE is at a maximum.

· KE is zero.

· Total energy is constant.

x (m)

E (J)ET

KE UE

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When the spring is passing through the equilibrium....


· EPE is zero.

· KE is at a maximum.


x (m)

E (J)ET

KE UE

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When the spring is all the way stretched....


· EPE is at a maximum.

· KE is zero.


x (m)

E (J)ET

KE UE

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7 At which location(s) is the kinetic energy of a mass-spring system a maximum?



7 At which location(s) is the kinetic energy of a mass-spring system a maximum?



Ans

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B

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8 At which location(s) is the spring potential energy (EPE) of a mass-spring system a maximum?


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8 At which location(s) is the spring potential energy (EPE) of a mass-spring system a maximum?



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D

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9 At which location(s) is the total energy of a mass-spring system a maximum?



9 At which location(s) is the total energy of a mass-spring system a maximum?



Ans

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E

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10 At which location(s) is the kinetic energy of a mass-spring system a minimum?



10 At which location(s) is the kinetic energy of a mass-spring system a minimum?



Ans

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D

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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?


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?


Ans

wer

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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?


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?


Ans

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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?

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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?


Ans

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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?


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?


Ans

wer

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SHM and UCM


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SHM and Circular Motion

There is a deep connection between Simple Harmonic Motion (SHM) and Uniform Circular Motion (UCM).

Simple Harmonic Motion can be thought of as a one-dimensional projection of Uniform Circular Motion.

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.

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Period

The time it takes for an object to complete one trip around a circular path is called its Period.

The symbol for Period is "T"

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,

page21svghttp://www.physics.uoguelph.ca/tutorials/shm/phase0.html

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15 If it takes 50 seconds for an object to travel around a circle 5 times, what is the period of its motion?


15 If it takes 50 seconds for an object to travel around a circle 5 times, what is the period of its motion?


Ans

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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?


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?


Ans

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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 seconds, what is the frequency (in Hz) of its motion?


Ans

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18 If an object is traveling in circular motion with a frequency of 7.0 Hz, how many revolutions will it make in 20s?


18 If an object is traveling in circular motion with a frequency of 7.0 Hz, how many revolutions will it make in 20s?


Ans

wer

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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

Also, recall from Uniform Circular Motion....

and

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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?

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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?


Ans

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In UCM, an object completes one circle, or cycle, in every T seconds. That means it returns to its starting position after T seconds.

In Simple Harmonic Motion, the object does not go in a circle, but it also returns to its starting position in T seconds.

SHM and Circular Motion

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 the frequency of the system?


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24 The period of a mass-spring system is 4.0s and the amplitude of its motion is 0.50m. How far does the mass travel in 4.0s?




Ans

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http://upload.wikimedia.org/wikipedia/commons/e/ea/Simple_Harmonic_Motion_Orbit.gif

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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)

· Frequency is the number of cycles completed per second (the same as frequency in UCM)

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Simple and Physical Pendulums


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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 Pendulumhas the form of if

But we learned before that

Substituting for k

Notice the "m" canceled out, the mass doesn't matter.

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26 What is the frequency of the pendulum of the previous slide (a length of 2.0m near the surface of the earth)?


26 What is the frequency of the pendulum of the previous slide (a length of 2.0m near the surface of the earth)?


Ans

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The Simple Pendulum

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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27 Which of the following factors affect the period of a pendulum?

A the acceleration due to gravityB the length of the stringC the mass of the pendulum bobD A & BE A & C


27 Which of the following factors affect the period of a pendulum?

A the acceleration due to gravityB the length of the stringC the mass of the pendulum bobD A & BE A & C


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D

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Energy in the Pendulum

The two types of energy in a pendulum are:

Gravitational Potential Energy

AND

The kinetic energy of the mass:

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The total mechanical energy is constant.

At any moment in time the total energy of the system is contant and comprised of those two forms.

Energy in the Pendulum

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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?


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?


Ans

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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)?


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)?


Ans

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Sinusoidal Nature of SHM


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The position as a function of for an object in simple harmonic motion can be derived from the equation:

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.

Position as a function of time

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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.

Position as a function of time

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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.


Velocity as a function of time

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We can also derive the equation for acceleration as a function of time.

Since a=rω2 can replace a with Aω2 as well as θ with ωt.

And again we can also replace ω with 2πf or 2π/T.


Acceleration as a function of time

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The Sinusoidal Nature of SHM

http://www.youtube.com/watch?v=eeYRkW8V7Vg&feature=PlayList&p=3AB590B4A4D71006&index=0

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.

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The Period and Sinusoidal Nature of SHM

x (displacement)

v (velocity)

a (acceleration)

Use this graph to answer the following questions.

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The Period and Sinusoidal Nature of SHM

x (displacement)

v (velocity)

a (acceleration)

T/4 T/2 3T/4 T

http://njc.tl/t8

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30 What is the acceleration when x = 0?

A a < 0B a = 0C a > 0D It varies.

T/4 T/2 3T/4 T

x (displacement)

v (velocity)

a (acceleration)

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31 What is the acceleration when x = A?


T/4 T/2 3T/4 T

x (displacement)

v (velocity)

a (acceleration)

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32 What is the acceleration when x = -A?


T/4 T/2 3T/4 T

x (displacement)

v (velocity)

a (acceleration)

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33 What is the velocity when x = 0?

A v < 0B v = 0C v > 0D A or C

T/4 T/2 3T/4 T

x (displacement)

v (velocity)

a (acceleration)

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34 What is the velocity when x = A?

A v < 0B v = 0C v > 0D A or C

T/4 T/2 3T/4 T

x (displacement)

v (velocity)

a (acceleration)

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35 Where is the mass when acceleration is at a maximum?

A x = AB x = 0C x = -AD A or C

T/4 T/2 3T/4 T

x (displacement)

v (velocity)

a (acceleration)

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36 Where is the mass when velocity is at a maximum?

A x = AB x = 0C x = -AD A or C

T/4 T/2 3T/4 T

x (displacement)

v (velocity)

a (acceleration)

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37 Which of the following represents the position as a function of time?

A x = 4 cos (2t)

B x = 2 cos (2t) C x = 2 sin (2t)

D x = 8 cos (2t)

T/4 T/2 3T/4 T

x (displacement)

v (velocity)

a (acceleration)a (acceleration)

v (velocity)

x (displacement)

T3T/4T/2T/4

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38 Which of the following represents the velocity as a function of time?

A v = -12 sin (2t)

B v = -12 cos (2t) C v = -4 cos (2t)

D v = -4 sin (2t)

T/4 T/2 3T/4 T

x (displacement)

v (velocity)

a (acceleration)

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39 Which of the following represents the acceleration as a function of time?

A v = -8 sin (2t)

B v = -8 cos (2t) C v = -4 cos (2t)

D v = -4 sin (2t)

T/4 T/2 3T/4 T

x (displacement)

v (velocity)

a (acceleration)

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AP Physics C - Mechanicscontent.njctl.org/courses/science/ap-physics-c-mechanics/... · 2015. 12. 10. · Slide 8 / 102 Hooke's Law For an object to be in simple harmonic motion,