8-1. Period of a simple harmonic oscillator

Transcription

1 8-1. Period of a simple harmonic oscillator 1. Purpose. Measure the period of a simple harmonic oscillator and compare it with the theoretical expectation. 2. Theory The oscillation period of a mass m attached to a spring is (1) where k is the spring constant. According to Hooke s law, the force applied on the mass is proportional to the deformation x of the spring (stretching or contraction), so it is expressed as F=kx. Hence, when we apply a force F to stretch the spring and measure the deformation x, we can obtain the spring constant by plotting F versus x and measuring the slope of the line passing through the points. 3. equipment

2 4. Method Measurement of the spring constant (1) Measure the mass m which includes the glider and the hook attached to it. (2) Put the air track in the horizontal position. Position the pulley at the end of the air track. (3) Connect the glider to the mass hanger with a string. Make the string pass through the pulley. (4) Attach the glider to the spring on the other side. Determine the equilibrium position of the glider. Hang different masses to the string and measure each time the distance by which the equilibrium position of the glider is modified and the spring is stretched. (5) Change the spring and repeat the above procedure. Measurement of the oscillation period (1) Pull the glider on one side by some fixed amount and release it. (2) Measure the period. (3) Add 50g to the glider, and repeat the above procedure. (4) Repeat after changing the spring.

3 5. Date sheet <Measurement of the spring constant>

4 <Measurement of the period> Theoretical calculation (1) Draw the graph of force versus elongation using the data in Table 1. Determine the spring constant by finding the slope of the curve. (2) Using the mass of the glider and the spring constant calculate the period according to equation (1) Calculate the error factor between the period measured experimentally and the theoretical expectation. 6. Questions (1) If a mass is added to the glider is the period faster of slower? (2) If the pulling distance is changed is the period also changed?

5 8-2. Motion on the slope 1. Purpose Change the angle of the slope and measure the oscillation period of a mass m, comparing it to the theoretical expectation. 2. Theory Refer to Experiment Equipment

6 4. Method Measurement of the theoretical period (1) Measure the mass m and write down it on [ 표 3]. (2) Set up the glider as shown on Fig. 3 [ 그림 2]. (3) Attach the spring to the glider and using the support jack one end of the track is inclined to increase its height. Hang different masses to the string and measure each time the distance by which the equilibrium position of the glider is modified and the spring is stretched. (4) Attach the glider to the spring on the other side. Turn on the air blower and remove friction of surface. According to the height of the track, the spring will be stretched. Maintain the angle of the track for the length of the spring to be smaller than half the length of the track. Measure the angle and write down it on [ 표3]. (5) Record equilibrium position in [ 표3] (6) Add a mass on the glider and record its new location. Prevent excessive stretching and repeat the experiment with five different mass Measurement of the experimental period (1) Pull the glider from the equilibrium position and make it vibrate. Record the time of fifth vibration. (2) Repeat it at least five times with same amplitude. (3) Change the angle of the air track and repeat the above procedure.

7 5. Result Theoretical determination of the oscillation period Experimental measurement of the oscillation period (1) Using the results in Table 3, calculate F = mg x sin θ. Draw the graph of F versus x and get the spring constant k. (2) Substitute the mass m and the spring constant in equation (1) and calculate the oscillation period. (3) Compare the two evaluations of the period obtained from Table 3 [ 표3] and Table 4 [ 표4]. <Questions> (1) If the tilting angle of the air track is changed does the oscillation period also change? (2) What happens to the oscillation period if the tilting angle is 90 degrees?

8 8-3. Springs in series and in parallel 1. Purpose Measure the oscillation period for springs in parallel and in series. Compare the results to the oscillation period for only one spring. 2. Theory The theoretical value of the oscillation period is given by: where m is the mass attached to the spring and k is the spring constant. As a result the spring constant is given by the following equation: Two springs can be attached to the same mass according to two different configurations: in parallel and in series. If the two springs have the same constant k, the effective spring constant of the system is given by: or by: depending on which configuration is chosen.

9 3. Equipment

10 4. Method Measurement of the spring constant (only one case) (1) Measure the mass of the glider. (2) Set up the glider as shown on Fig. 3 [ 그림 3]. (3) Let it oscillate and measure the period. (4) Connect the glider to two springs in series as shown in Fig. 4 [ 그림 4] and repeat (3). (5) Connect the glider to two springs in parallel as shown in Fig. 5 [ 그림 5] and repeat (3). (6) Connect the glider to two springs on both sides as shown in Fig. 6 [ 그림 6] and repeat (3).

11 5. Results <Questions> (1) Which one of the two configurations (parallel or series) is consistent with? (2) Which one of the two configurations (parallel or series) is consistent with? (3) Is the way to connect the springs to both sides of the glider consistent with the configuration in parallel? Or is it consistent to the configuration in series? 8-4. Coupled harmonic vibration 1. Purpose Measure the oscillation period of two gliders that are connected with 3 springs, so that there are two degrees of freedom 2. Theory

12 If we consider two gliders connected to 3 springs as shown in Fig.7 [ 그림 7], the equation of motion is given by: and the two masses oscillate with different frequencies. However, in two special cases both masses oscillate with the same frequency. This happens when (anti-symmetric vibration) and when (symmetric vibration). In each case the two masses have a different common oscillation frequency. If the two masses are connected weakly, i.e., we can use a linear approximation of the equations of motion for x 1 and x 2 in terms of the small parameter ε = k 3 /2k: 3. Equipment

13 4. Method (1) Put two gliders on the air track. Put the air track in the horizontal position. (2) Attach the 3 springs to the gliders and set up the coupled harmonic oscillator. (3) Measure the oscillation period corresponding to each of the cases explained above. Symmetric vibration (4) As shown in Fig.8 [ 그림 8], move the two gliders by the same distance and in the same direction from their equilibrium points and turn on the air blower. When you let them oscillate they start symmetric vibration Anti - Symmetric vibration (5) As shown in Fig.9 [ 그림 9], move the two gliders by same distance and in opposite directions away from their equilibrium points, and turn on the air blower. When you

14 let them oscillate they start anti-symmetric vibration Coupled harmonic vibration (6) As shown in Fig. 10 [ 그림 10], move only one glider by an arbitrary distance and in an arbitrary direction from its equilibrium point, and turn on the air blower. When the glider is released both gliders start coupled harmonic vibration.