# Harmonic Motion

1) Use linear least squares to find the slope of this force vs. position data. From this slope, calculate the effective spring constant k:

2) Calculate the natural frequency f0 you should expect for this mechanical oscillator:

3) Using the time difference between the first and last peaks, calculate the frequency of the oscillation:

4) Consider Equation 9, page 135. Perform a linear least squares fit on ln(A) vs.t to get alpha :

5) 5) Use the alpha from question 4 along with the mass of the glider, m, to get the damping coefficient b (the coefficient of the velocity term in the equation of motion, defined by Equation 7 shown on page 135):

6) Use the undamped frequency ( w0), the mass of the glider (m), and the damping coefficient (b) to predict Q:

7) If the amplitude of the system driven at a frequency MUCH lower than the resonant frequency is 0.0346 m, predict the amplitude of the system when the driving frequency is raised to the resonant frequency:

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In Lab 11, you will study harmonic motion. You will be measuring the position as a function of time of a glider attached by springs to the ends of an air track under various conditions. (We use an air track because these are very friction sensitive studies and the air track provides a cushion of air for the glider to ride on, resulting in exceptionally low friction between the glider and the track). The following exercises are intended to help you become familiar with some of the concepts and techniques that you will be using in your experiments. The numerical values involved in the exercises are similar to ones you will actually measure, so you can already have some feeling for what to expect.

Investigation 1 of Lab 11 tests the predicted relation of the natural frequency (f0) to the physical parameters of the system: the mass of the glider (m) and the effective spring constant (k). With a balance you measure the mass of the glider to be m = 0.312 kg. (We ignore the mass of the springs in this lab). To determine k, consider the setup shown in the figure on page 133 of the manual. The procedure is to measure the position x of the glider as force F is applied to the glider (the force probe both applies the force that stretches the spring and measures it). In the actual investigation you will be using Data Studio to create a graph, then fit a straight line to the graph. Here we provide you with a table:

d (m) F (N)

0.4290 0.212

0.4681 0.434

0.5074 0.657

0.5466 0.879

1) Use linear least squares to find the slope of this force vs. position data. From this slope, calculate the effective spring constant k:

2) Calculate the natural frequency f0 you should expect for this mechanical oscillator:

Investigation 2 concerns the free oscillations of a damped harmonic oscillator (refer to the figure on page 134). The setup is exactly the same as for Investigation 1 with the exception that some magnets are placed on the skirts of the glider to provide eddy current damping.

IMPORTANT NOTE: These magnets were attached to the glider in Investigation 1, but they were positioned so as to NOT generate any damping. This means that the mass of the glider is unchanged during the entire lab.

You pull the glider to one side from its equilibrium position, release it from rest, and record the position vs. time of the decaying oscillation. Your data will be somewhat like that shown in Figure 2 on page 137 of the manual. Using the SmartTool, the following peak times and amplitudes were measured:

Peak t (s) A (m)

1 0.0147 0.3452

2 1.4895 0.2756

3 2.9644 0.2201

4 4.4392 0.1757

3) Using the time difference between the first and last peaks, calculate the frequency of the oscillation:

We could try to use the calculated frequency of the undamped system and the measured frequency of the damped system to get the damping term , but this frequency shift is very small and it is difficult to measure the "peak times" with great precision. A better way to calculate is to look at the decay of the peak amplitudes over time.

4) Consider Equation 9, page 135. Perform a linear least squares fit on ln(A) vs.t to get alpha :

5) Use the alpha from question 4 along with the mass of the glider, m, to get the damping coefficient b (the coefficient of the velocity term in the equation of motion, defined by Equation 7 shown on page 135):

In Investigation 3, you measure the amplitude A of oscillation as a function of frequency for the driven oscillator (refer again to figure on page 134). The spring support at one end of the air track is oscillated at frequency f and the amplitude A(f) of the motion of the glider is recorded as a function of frequency. Figure 3 on page 143 shows data as a ratio A/A0 for various values of Q (defined by Equation 38), where A0 is the amplitude measured at a frequency very small compared to the natural frequency f0.

6) Use the undamped frequency ( w0), the mass of the glider (m), and the damping coefficient (b) to predict Q:

7) If the amplitude of the system driven at a frequency MUCH lower than the resonant frequency is 0.0346 m, predict the amplitude of the system when the driving frequency is raised to the resonant frequency:

... (SEE ATTACHED)

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#### Solution Summary

Calculates effective spring constant, natural frequency, frequency of the oscillation, alpha, damping coefficient, amplitude

Nine review problems: Force, energy, motion, frequency, amplitude, harmonic motion, oscillation, speed, tuning fork, beat frequency

See attachment for proper format.

1. An archer pulls his bow string back 0.4 m by exerting a force on the string that increases uniformly from zero to 230 N. What is the equivalent force constant of the bow? How much work does the archer do in pulling the bow?

2. A ball dropped from a height of h makes a perfectly elastic collision with the ground. Assuming no energy is lost due to air resistance, show that the ensuing motion is periodic and determine its period.

3. The position of a particle is given by the expression x=4cos(3?t+?), where x is in meters and t is in seconds. Determine the frequency and period of the motion, the amplitude of the motion, the phase constant, and the position of the particle after first 0.25 seconds of motion. Derive an expressions for particle velocity and acceleration.

4. A piston in a gasoline engine is in simple harmonic motion. If the extremes of its displacement away from its center point are ±5.0 cm, derive the equation of this motion when engine is running at the rate of 3600 rev/min.

5. After a thrilling plunge, bungee-jumpers bounce freely on the bungee cord through many cycles. You can make a pest by figuring out the mass of each person, using a proportion which you set up by solving this problem: An object of mass M is oscillating freely on a vertical spring with a period T. The object of unknown mass m on the same spring oscillates with a period t. Determine the force constant (spring stiffness k) and the unknown mass.

6. A stone is dropped from the top of a cliff. The splash it makes when striking the water below is heard 3.5 s later. How high is the cliff? (Assume speed of sound equal to 340 m/s)

7. At the Indianapolis 500, you can measure the speed of cars just by listening to the difference in pitch of the engine noise between approaching and receding cars. Suppose the sound of a certain car drops by a full octave (262 Hz) as it goes by on the straightway. How fast is it going?

8. A tuning fork is set into vibration above a vertical open tube filled with water. The water level is allowed to drop slowly. As it does so, the air in the tube above the water level is heard to resonate with the tuning fork when the distance from the tube opening to the water level is 0.125 m and again at 0.395 m. What is the frequency of the tuning fork?

9. In certain ranges of a piano keyboard, more than one string is tuned to the same note to provide extra loudness. For example, the note at 110 Hz has two strings at this frequency. If one string slips from its normal tension of 600 N to 540 N, what beat frequency is heard when the hammer strikes the two strings simultaneously?

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