© 2010 Pearson Education, Inc. Slide 14-2 14 Oscillations

© 2010 Pearson Education, Inc. Slide 14-2

14 Oscillations




© 2010 Pearson Education, Inc.

Equilibrium and Oscillation

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Linear Restoring Forces and Simple Harmonic Motion

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Linear Restoring Forces and Simple Harmonic Motion

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For a hanging mass at equilibrium

The weight is balanced by the force from the spring

From this one can readily measure and calculate the spring constant

And a theoretical value for the frequency can be calculated using the previous equations


Frequency and PeriodThe frequency of oscillation depends on physical properties of the oscillator; it does not depend on the amplitude of the oscillation.

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Snap Quiz! 1. The type of function that describes simple harmonic motion is

A. linearB. exponentialC. quadraticD. sinusoidalE. inverse

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Answer 1. The type of function that describes simple harmonic motion is

A. linearB. exponentialC. quadraticD. sinusoidalE. inverse

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Energy in Simple Harmonic MotionAs a mass on a spring goes through its cycle of oscillation, energy is transformed from potential to kinetic and back to potential.

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Energy in Simple Harmonic MotionThe total energy is equal to the potential energy when x = A, and is equal to the kinetic energy when x = 0. It is also equal to the sum of kinetic and potential energies at any time.

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In the equilibrium position, potential energy = 0 and

At maximum displacement locations, x = A and Ek = 0 and


Sinusoidal Relationships

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Mathematical Description of Simple Harmonic Motion

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The position, velocity, and acceleration at any given time can be found from these equations given the amplitude, A and the frequency


A pendulum is pulled to the side and released. Rank the following positions in terms of the speed, from highest to lowest. There may be ties.

Additional Questions

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Damping

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Resonance

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Four different masses are hung from four springs with unstretched length 10 cm, causing the springs to stretch as noted in the following diagram:

Now, each of the masses is lifted a small distance, released, and allowed to oscillate. Rank the oscillation frequencies, from highest to lowest.

A. a b c dB. d c b aC. a b c d


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Four different masses are hung from four springs with unstretched length 10 cm, causing the springs to stretch as noted in the following diagram:



Answer

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Four 100 g masses are hung from four springs, each with unstretched length 10 cm. The four springs stretch as noted in the following diagram:




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Four 100 g masses are hung from four springs, each with unstretched length 10 cm. The four springs stretch as noted in the following diagram:



Answer

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Summary

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Summary

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Pendulum Motion Demonstrator on Youtube

The demonstration at

http://www.youtube.com/watch?v=yVkdfJ9PkRQ

The lengths of the cords increase in uniform increments

All of the motions you see are the result of combinations of the frequencies of each of the pendulums


Play with the PHET pendulum demonstrator

The PHET pendulum demonstrator allows you to change pendulum mass, length, and the acceleration of gravity

Two pendulums can be run simultaneously

http://phet.colorado.edu/en/simulation/pendulum-lab


A typical earthquake produces vertical oscillations of the earth. Suppose a particular quake oscillates the ground at a frequency 0.15 Hz. As the earth moves up and down, what time elapses between the highest point of the motion and the lowest point?

A. 1 sB. 3.3 sC. 6.7 sD. 13 s


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Answer A typical earthquake produces vertical oscillations of the earth. Suppose a particular quake oscillates the ground at a frequency 0.15 Hz. As the earth moves up and down, what time elapses between the highest point of the motion and the lowest point?

A. 1 sB. 3.3 sC. 6.7 sD. 13 s

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Additional Example ProblemWalter has a summer job babysitting an 18 kg youngster. He takes his young charge to the playground, where the boy immediately runs to the swings. The seat of the swing the boy chooses hangs down 2.5 m below the top bar. “Push me,” the boy shouts, and Walter obliges. He gives the boy one small shove for each period of the swing, in order keep him going. Walter earns $6 per hour. While pushing, he has time for his mind to wander, so he decides to compute how much he is paid per push. How much does Walter earn for each push of the swing?

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A 500 g block is attached to a spring on a frictionless horizontal surface. The block is pulled to stretch the spring by 10 cm, then gently released. A short time later, as the block passes through the equilibrium position, its speed is 1.0 m/s. What is the block’s period of oscillation? What is the block’s speed at the point where the spring is compressed by 5.0 cm?

Additional Example Problems

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Reading Quiz 2. A mass is bobbing up and down on a spring. If you increase

the amplitude of the motion, how does this affect the time for one oscillation?

A. The time increases.B. The time decreases.C. The time does not change.

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Answer2. A mass is bobbing up and down on a spring. If you increase

the amplitude of the motion, how does this affect the time for one oscillation?

A. The time increases.B. The time decreases.C. The time does not change.

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Reading Quiz 3. If you drive an oscillator, it will have the largest amplitude if you

drive it at its _______ frequency.

A. specialB. positiveC. resonantD. dampedE. pendulum

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Answer 3. If you drive an oscillator, it will have the largest amplitude if you

drive it at its _______ frequency.

A. specialB. positiveC. resonantD. dampedE. pendulum

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Checking UnderstandingA set of springs all have initial length 10 cm. Each spring now has a mass suspended from its end, and the different springs stretch as shown below.

Now, each mass is pulled down by an additional 1 cm and released, so that it oscillates up and down. Rank the frequencies of the oscillating systems A, B, C and D, from highest to lowest.

A. B D C AB. B A D CC. C A D B D. A C B D

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AnswerA set of springs all have initial length 10 cm. Each spring now has a mass suspended from its end, and the different springs stretch as shown below.

Now, each mass is pulled down by an additional 1 cm and released, so that it oscillates up and down. Rank the frequencies of the oscillating systems A, B, C and D, from highest to lowest.

A. B D C AB. B A D CC. C A D B D. A C B D

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A series of pendulums with different length strings and different masses is shown below. Each pendulum is pulled to the side by the same (small) angle, the pendulums are released, and they begin to swing from side to side.

Rank the frequencies of the five pendulums, from highest to lowest.

A. A E B D CB. D A C B EC. A B C D E D. B E C A D

Checking Understanding

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A series of pendulums with different length strings and different masses is shown below. Each pendulum is pulled to the side by the same (small) angle, the pendulums are released, and they begin to swing from side to side.

Rank the frequencies of the five pendulums, from highest to lowest.

A. A E B D CB. D A C B EC. A B C D E D. B E C A D

Answer

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Example Problems

The first astronauts to visit Mars are each allowed to take along some personal items to remind them of home. One astronaut takes along a grandfather clock, which, on earth, has a pendulum that takes 1 second per swing, each swing corresponding to one tick of the clock.

When the clock is set up on Mars, will it run fast or slow?

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Example

A 5.0 kg mass is suspended from a spring. Pulling the mass down by an additional 10 cm takes a force of 20 N.

If the mass is then released, it will rise up and then come back down. How long will it take for the mass to return to its starting point 10 cm below its equilibrium position?

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A ball on a spring is pulled down and then released. Its subsequent motion appears as follows:

1. At which of the above times is the displacement zero?2. At which of the above times is the velocity zero?3. At which of the above times is the acceleration zero?4. At which of the above times is the kinetic energy a maximum?5. At which of the above times is the potential energy a maximum?6. At which of the above times is kinetic energy being transformed to

potential energy?7. At which of the above times is potential energy being transformed

to kinetic energy?

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Example Problem


A pendulum is pulled to the side and released. Its subsequent motion appears as follows:

1. At which of the above times is the displacement zero?2. At which of the above times is the velocity zero?3. At which of the above times is the acceleration zero?4. At which of the above times is the kinetic energy a maximum?5. At which of the above times is the potential energy a maximum?6. At which of the above times is kinetic energy being transformed to

potential energy?7. At which of the above times is potential energy being transformed

to kinetic energy?

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Example Problem


A 204 g block is suspended from a vertical spring, causing the spring to stretch by 20 cm. The block is then pulled down an additional 10 cm and released. What is the speed of the block when it is 5.0 cm above the equilibrium position?

Example Problem

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© 2010 Pearson Education, Inc. Slide 14-2 14 Oscillations