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Reading Quiz The type of function that describes simple harmonic motion is linear exponential quadratic sinusoidal inverse Answer: D These reading quiz questions are a nice way to get the students minds into the game at the start of class. They are fairly straight forward.

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**Answer The type of function that describes simple harmonic motion is**

linear exponential quadratic sinusoidal inverse Answer: D

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Reading Quiz 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? The time increases. The time decreases. The time does not change. Answer: C

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Answer 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? The time increases. The time decreases. The time does not change. Answer: C

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Reading Quiz If you drive an oscillator, it will have the largest amplitude if you drive it at its _______ frequency. special positive natural damped pendulum Answer: C

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Answer If you drive an oscillator, it will have the largest amplitude if you drive it at its _______ frequency. special positive natural damped pendulum Answer: C

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**Equilibrium and Oscillation**

I used this to start a discussion about if an EKG is Simple Harmonic Motion (SHM). First I asked if it could be and let the students discuss with their neighbor. Then we had a whole class discussion. It is not because there’s not a linear restoring force that causes the heart to beat. You can see this because it’s not a sinusoidal function. The ball in a bowl is however, a nice example.

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**Linear Restoring Forces and Simple Harmonic Motion**

Simple harmonic motion (SHM) is the motion of an object subject to a force that is proportional to the object's displacement. One example of SHM is the motion of a mass attached to a spring. In this case, the relationship between the spring force and the displacement is given by Hooke's Law, F = -kx, where k is the spring constant, x is the displacement from the equilibrium length of the spring, and the minus sign indicates that the force opposes the displacement.

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**Linear Restoring Forces and Simple Harmonic Motion**

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

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**Sinusoidal Relationships**

This page demonstrates how the position equation can be sin or cos. It simply depends on the initial amplitude which means it just depends on when you start time.

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**Each dimension of circular motion is a sinusoidal funtion.**

Animation of sine and cosine

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**Mathematical Description of Simple Harmonic Motion**

Students have a surprisingly hard time with many aspects of the math of Acos(2pift) The next series of questions helps with some of it. I also used the workbooks from Knight, Jones and Fields in recitation and the work for this chapter really helped students sort out how the math works here.

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**What is the largest value cos(2pft) can ever have?**

1 2pft

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**What is the largest value cos(2pft) can ever have?**

1 2pft

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**What is the largest value sin(2pft) can ever have?**

1 2pft

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**What is the largest value sin(2pft) can ever have?**

1 2pft

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**Maximum displacement, velocity and acceleration**

What is the largest displacement possible? A 2pft ½ A

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**Maximum displacement, velocity and acceleration**

What is the largest displacement possible? A 2pft ½ A

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**Maximum displacement, velocity and acceleration**

What is the largest velocity possible? 2pf 2Apf Apf

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**Maximum displacement, velocity and acceleration**

What is the largest velocity possible? 2pf 2Apf Apf

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**Maximum displacement, velocity and acceleration**

What is the largest acceleration possible? 2A(pf)2 4p2Af2 2Apf

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**Maximum displacement, velocity and acceleration**

What is the largest acceleration possible? 2A(pf)2 4p2Af2 2Apf

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**Energy in Simple Harmonic Motion**

As 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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Solving Problems

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Equations

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**Additional Example Problem**

Walter 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? I let the class try this problem out while the equations were showing. Then after they had worked for a reasonable amount of time I got ideas from them how to solve the problem and then did it on the board.

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**Additional Example Problem**

Walter 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? T = 2p √(l/g) = 3.17 s 3600 s/hr (1 push/3.17s) = 1136 pushes/ hr $6/hr (1hr/1136 pushes) = $ / push Or half a penny per push!

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**Checking Understanding**

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. 25 Rank the frequencies of the five pendulums, from highest to lowest. A E B D C D A C B E A B C D E B E C A D Answer: A

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Answer 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. 25 Rank the frequencies of the five pendulums, from highest to lowest. A E B D C D A C B E A B C D E B E C A D Answer: A f = 1/2p √(g/l) Frequency is proportional to the inverse of the length. So longer has a lower frequency

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Example Problem We think of butterflies and moths as gently fluttering their wings, but this is not always the case. Tomato hornworms turn into remarkable moths called hawkmoths whose flight resembles that of a hummingbird. To a good approximation, the wings move with simple harmonic motion with a very high frequency—about 26 Hz, a high enough frequency to generate an audible tone. The tips of the wings move up and down by about 5.0 cm from their central position during one cycle. Given these numbers, What is the maximum velocity of the tip of a hawkmoth wing? What is the maximum acceleration of the tip of a hawkmoth wing? Again, let the class work on this in small groups and then did on the board.

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Example Problem We think of butterflies and moths as gently fluttering their wings, but this is not always the case. Tomato hornworms turn into remarkable moths called hawkmoths whose flight resembles that of a hummingbird. To a good approximation, the wings move with simple harmonic motion with a very high frequency—about 26 Hz, a high enough frequency to generate an audible tone. The tips of the wings move up and down by about 5.0 cm from their central position during one cycle. Given these numbers, What is the maximum velocity of the tip of a hawkmoth wing? What is the maximum acceleration of the tip of a hawkmoth wing? vmax = 2pfA = 2 p 26Hz 0.05m = 8.17 m/s amax = A (2pf)2 = (2 p 26Hz) m = 1334 m/s2

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Example Problem A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is the displacement zero? At which of the above times is the velocity zero? At which of the above times is the acceleration zero? At which of the above times is the kinetic energy a maximum? At which of the above times is the potential energy a maximum? At which of the above times is kinetic energy being transformed to potential energy? At which of the above times is potential energy being transformed to kinetic energy? Students have a hard time with several of these. The next series of slides breaks each into its own question. It took a good 15 minutes or more of class time to work through the ideas in their groups and as a whole class but it seemed to work well. Answer: A, E, I 2) C, G 3) A, E, I 4) A, E, I 5) C, G 6) B, F 7) D, H

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**Example Problem At which of the above times is the displacement zero?**

A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is the displacement zero? C, G A, E, I B, F D, H

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**Example Problem At which of the above times is the displacement zero?**

A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is the displacement zero? C, G A, E, I B, F D, H

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**Example Problem At which of the above times is the velocity zero? C, G**

A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is the velocity zero? C, G A, E, I B, F D, H Answer: A, E, I 2) C, G 3) A, E, I 4) A, E, I 5) C, G 6) B, F 7) D, H

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**Example Problem At which of the above times is the velocity zero? C, G**

A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is the velocity zero? C, G A, E, I B, F D, H

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**Example Problem At which of the above times is the acceleration zero?**

A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is the acceleration zero? C, G A, E, I B, F D, H This one gave them the most difficulty so we looked at the free body diagram for each (on the next slide) to make sense out of it.

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**Example Problem Without ball**

A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: Without ball This one gave them the most difficulty so we looked at the free body diagram for each to make sense out of it. At which of the above times is the acceleration zero? C, G A, E, I B, F D, H

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Example Problem A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is the kinetic energy a maximum? C, G A, E, I B, F D, H Answer: A, E, I 2) C, G 3) A, E, I 4) A, E, I 5) C, G 6) B, F 7) D, H

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Example Problem A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is the kinetic energy a maximum? C, G A, E, I B, F D, H

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Example Problem A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is the potential energy a maximum? C, G A, E, I B, F D, H

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Example Problem A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is the potential energy a maximum? C, G A, E, I B, F D, H

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Example Problem A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is kinetic energy being transformed to potential energy? C, G A, E, I B, F D, H

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Example Problem A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is kinetic energy being transformed to potential energy? C, G A, E, I B, F D, H

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Example Problem A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is potential energy being transformed to kinetic energy? C, G A, E, I B, F D, H

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Example Problem A ball on a spring is pulled down and then released. Its subsequent motion appears as follows: At which of the above times is potential energy being transformed to kinetic energy? C, G A, E, I B, F D, H

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Example Problem A pendulum is pulled to the side and released. Its subsequent motion appears as follows: At which of the above times is the displacement zero? At which of the above times is the velocity zero? At which of the above times is the acceleration zero? At which of the above times is the kinetic energy a maximum? At which of the above times is the potential energy a maximum? At which of the above times is kinetic energy being transformed to potential energy? At which of the above times is potential energy being transformed to kinetic energy? Answer: C, G 2) A, E, I 3) C, G 4) C, G 5) A, E, I 6) D, H 7) B, F

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Example Problem A pendulum is pulled to the side and released. Its subsequent motion appears as follows: At which of the above times is the displacement zero? C, G At which of the above times is the velocity zero? A, E, I At which of the above times is the acceleration zero? C, G At which of the above times is the kinetic energy a maximum? C, G At which of the above times is the potential energy a maximum? A, E, I At which of the above times is kinetic energy being transformed to potential energy? D, H At which of the above times is potential energy being transformed to kinetic energy? B, F Answer: C, G 2) A, E, I 3) C, G 4) C, G 5) A, E, I 6) D, H 7) B, F

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Great video showing how the bottom of a slinky does not fall until the compression has relaxed – very counterintuitive!!

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Prediction Why is the slinky stretched more at the top than the bottom? The slinky is probably not in good shape and has been overstretched at the top. The slinky could be designed that way and if he holds it the other way around, it’ll be denser at the top than the bottom. There’s more mass pulling on the top loop so it has to stretch more to have a upward force kx equal to the weight mg.

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Prediction Why is the slinky stretched more at the top than the bottom? The slinky is probably not in good shape and has been overstretched at the top. The slinky could be designed that way and if he holds it the other way around, it’ll be denser at the top than the bottom. There’s more mass pulling on the top loop so it has to stretch more to have an upward force kx equal to the weight mg.

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Prediction How will the slinky fall? A. The entire thing will fall to the ground w/ an acceleration of g. B. The center of mass will fall w/ an acceleration of g as the slinky compresses making it look like the bottom doesn’t fall as fast. C. The bottom will hover stationary in the air as the top falls until the top hits the bottom. D. Other watch video

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Prediction How will the slinky fall? A. The entire thing will fall to the ground w/ an acceleration of g. B. The center of mass will fall w/ an acceleration of g as the slinky compresses making it look like the bottom doesn’t fall as fast. C. The bottom will hover stationary in the air as the top falls until the top hits the bottom. D. Other

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Why does this happen?

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Example Problem 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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Example Problem A car rides on four wheels that are connected to the body of the car by springs that allow the car to move up and down as the wheels go over bumps and dips in the road. Each spring supports approximately 1/4 the mass of the vehicle. A lightweight car has a mass of 2400 lbs. When a 160-lb person sits on the left front fender, this corner of the car dips by about 1/2 in. What is the spring constant of this spring? When four people of this mass are in the car, what is the oscillation frequency of the vehicle on the springs?

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Example Problem Manufacturers are now making shoes with springs in the soles. A spring in the heel will decrease the impact force when your heel strikes, but, equally important, it can store energy that is returned when the foot rolls forward to push off. Ideally, a shoe would be designed so that, when your heel strikes, the spring compresses and then rebounds at a natural point in your stride. We can model this rebound as a segment of an oscillation to get some idea of how such a shoe should be designed. In a moderate running gait, your heel is in contact with the ground for 0.1 s or so. For optimal timing, what should the period of the motion for the mass (the runner) and spring (the spring in the shoe) system be? What should be the spring constant of the spring for a 70-kg man? When the man puts all his weight on the heel, how much should the spring compress?

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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?

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