Pendulum Experiment: Effect of Mass, String Length, and Amplitude on Period, Study notes of Law

Download Pendulum Experiment: Effect of Mass, String Length, and Amplitude on Period and more Study notes Law in PDF only on Docsity!

Contents

1 PENDULUM EXPERIMENT 3

2 SIMPLE HARMONIC MOTION 9

3 STANDING WAVES ON STRINGS 15

4 STANDING WAVES IN AIR COLUMNS 21

5 SOUND INTENSITY - THE DECIBEL SCALE 27

6 REVERBERATION TIME 31

2 CONTENTS

4 EXPERIMENT 1. PENDULUM EXPERIMENT

Now you are ready to make observations. It is best to work in teams, with one person counting the number of swings while the other measures the time using a stop-watch or a smartphone. Determine the time required for 25 complete swings (or any comparable number of swings, it is up to you). Do this three times for each particular set of variables, and record your measurements on the data sheets.

VARYING MASS

Set the length at some convenient value (say 50 cm), and measure the time for 25 swings, for four different masses starting with 20 g or less, and increasing to 500 g or more. Try to start the pendulum with the same amplitude in each case say 10o.

Length:− − − − − − −cm.

time for 25 swings mean / 25 mass (g) trial 1 (s) trial 2 (s) trial 3 (s) mean* (s) period (s)

• The mean is just the average of your three time measurements, for a given mass.

VARYING LENGTH

Keep the mass fixed (at 100 or 200 g), and measure the time for 25 swings, for four different lengths, starting at about 10 cm, and increasing to the largest length the setup allows.Try to start the pendulum with same amplitude in each case, say 10o.

Mass:− − − − − − −g.

VARYING THE AMPLITUDE

Keep the mass fixed (at 100 or 200 g), and set the length at 50 cm or so. Measure the time for 25 swings, for four different amplitudes, starting at about 10o, and increasing to 90o.

Mass:− − − − − − −g; Length:− − − − − − −cm. time for 25 swings mean / 25 amplitude trial 1 (s) trial 2 (s) trial 3 (s) mean (s) period (s)

ANALYSIS

1. Compute the mean times and record in the table. If one of your three time measurements is very different from the other two, notify the instructor

• there may be an error, and it may be advisable to repeat that measurement.

2. Based on your measurements, state the effect of varying mass on the period. Do your measurements support or refute your original hypothesis?
3. Based on your measurements, state the effect of varying length on the period. Do your measurements support or refute your original hypothesis?
4. Make a plot of the period versus the amplitude. Based on your mea- surements, state the effect of varying amplitude on the period. Do your measurements support or refute your original hypothesis?

8 EXPERIMENT 1. PENDULUM EXPERIMENT

10 EXPERIMENT 2. SIMPLE HARMONIC MOTION

THEORY

When a body of mass m is suspended from a spring, its weight (weight=F = mg) causes the spring to elongate. The elongation x, is directly proportional to the force exerted F = kx,

where k is the spring constant (or force constant). k measures the stiffness of the spring. The above relationship is known as Hooke’s law and applies to all elastic materials within the elasticity limit. According to Newton’s third law, the force with which the spring is acting on the suspended mass, is the opposite of the above, F = −kx.

The sign minus in this formula reflects the fact that the force from the spring on the mass is a restoring force. Restoring force is the main prerequisite for oscillations. If the body is pulled down and then released, it will oscillate about the equilibrium position (the position of the body when the spring was station- ary). This motion is called simple harmonic motion. In this experiment we will measure the period of oscillation and see how amplitude and mass affect the period. The theoretical value for the period, T , of the motion is given by

T = 2π

M

k

where M is the effective mass of the vibrating system, which is made up of the mass of the suspended body plus a part of the mass of the spring, since the spring itself is also vibrating. From mechanics follows

M = m +

mspring 3

EXPERIMENT

Part A. To determine how the period of oscillations depends on the ampli- tude. A1. Suspend a 100 g mass from the spring.

A2. Make careful measurements of T (period) versus A (amplitude) for various amplitudes such as 1.5 cm, 3 cm and 4.5 cm. (To get an amplitude of 3 cm, displace the mass 3 cm from its rest position and then let it go.) For each amplitude, measure the time for 50 complete cycles. Fill Table A with data. (The values above are just for your orientation. You may choose modified values.)

TABLE A

Suspended mass, m =

Amplitude Time for 50 cycles Period Frequency A (cm) t (s) T = t/50 (s) f = 1/T (Hz)

Part B. To find the mass of the spring and the spring constant k.

B1. Use the balance to determine the mass of the spring and of the weight holder.

B2. Suspend the holder and move the scale vertically so the pointer is on zero. Add 50 g and allow the system to come to equilibrium. Record the added mass and the new position of the pointer in Table B. Continue in this manner, adding 50 g at a time until the final load is 250 g.

TABLE B

ANALYSIS

Part A.

1. Using your data from A2 calculate the period and frequency of vibra- tion for each observation and fill in Table A.
2. Does the period of vibration depend on the amplitude for the cases you studied?

Part B.

3. From the data in Table B, plot a graph of F = mg versus displacement x, where g = 9.8 m/s^2. Make sure that the mass is expressed in kg (1000 g = 1 kg) and the displacement x is measured in meters (100 cm = 1 m).
4. Determine the spring constant k (in N/m) from the slope of this graph, as explained in Lab 1.

Part C.

5. Calculate the theoretical value for the period, T , for the masses used in measurement of the period, using the formulas (1) and (2), k being the value determined from your graph above.
6. Compare your experimental values for T in Table C with the theoretical values for T by calculating the percentage discrepancy. Fill in the Table below.

Effective mass, M Period (experiment) Period (theory) % discrepancy M (g) Texp (s) Ttheor (s)

7. Discuss your results and discrepancies. Did the lab produce the ex- pected outcome?

14 EXPERIMENT 2. SIMPLE HARMONIC MOTION

16 EXPERIMENT 3. STANDING WAVES ON STRINGS

THEORY

Standing waves are produced by the interference of two waves of the same wavelength, speed of propagation and amplitude, travelling in opposite di- rections through the same medium.

If one end of a light, flexible string is attached to a vibrator and the other end passes over a fixed pulley to a weight holder, the waves travel down the string to the pulley and are then reflected, producing a reflected wave moving in the opposite direction. The vibration of the string is then a composite motion resulting from the combined effect of the two oppositely directed waves. If the proper relationship exists between the frequency, the length and the tension, a standing wave is produced and when the conditions are such as to make the amplitude of the standing wave a maximum, the system is said to be in resonance. A standing wave has points of zero displacement (due to destructive interference) and points of maximum displacement (due to constructive interference). The positions of no vibration are called nodes (N) and the positions of maximum vibration are called antinodes (A). The segment between two nodes is called a loop.

Standing waves with one, two, three and four loops are given below.

The solid line represents the form of the string at an instant of maximum displacement and the dotted line represents the configuration one half-period later when the displacements are reversed. In each case λ = 2l, where λ is the wavelength and l is the distance between two nearby nodes. For a string

with both ends fixed the allowed wavelengths for standing waves can only take fixed values related to the length L of the string, as can be seen in the figure above. If one changes the tension in a vibrating string, the number of loops between the ends of the string change. As a result the distance between neighboring nodes changes, thus producing a change in wavelength. The speed of the wave can be obtained if the frequency f is known and the wave length λ has been measured:

v = λf. (3.1)

The frequency is fixed by the string vibrator; the wavelength can only take on fixed values related to the length of the string, as shown in the figure above. Thus, standing waves can only exist for particular values of v that is controlled in this experiment by the tension of the string. The velocity of the wave is given by the Mercenne’s law:

v =

T

m

where m is the mass per length of the string and T is the tension. The tension of the string (in newtons) equals the total hanging mass M times the gravitational acceleration g = 9. 8 m/s^2 , that is, T = M g. The main objective of the work is to compare the experimental value of the wave speed given by Eq. (1) and its theoretical value of Eq. (2).

EXPERIMENTAL SETUP

A string is attached to a vibrator made of steel and then passed over a small pulley as shown in Fig.1. The coil producing the alternating magnetic field acting on the vibrator is being fed by the standard ac current with a fixed frequency of 60 Hz. The attraction force exerted on the vibrator is proportional to the square of the magnetic field and thus of the electric current in the cirquit. As the result, the vibrator is vibrating at the double frequency, 120 Hz. The weight on the hanger has to be adjusted to achieve the tension T and thus the wave speed v at which a standing wave is clearly visible. As the range of the tension is limited, not all kinds of standing waves can be

Number λ vexp = f λ M T = M g vtheor =

T /m % discr of loops (m) (m/s) (kg) (N) (m/s)

1 loops

2 loops

3 loops

4 loops

ANALYSIS

1. Calculate the mass per unit length and express it in kg/m.
2. Calculate the velocity of the wave on the string using equations (1) and (2) and compare the results. Calculate the percent discrepancy. Fill all of this in the data table.
3. In your conclusion discuss your results.
4. If the frequency of the vibrator were 240 Hz, calculate theoretically how much tension is necessary to produce a standing wave of two loops. Consider that the string is fixed at its two ends, has a length of 1 meter and has the same mass per length as determined before.

20 EXPERIMENT 3. STANDING WAVES ON STRINGS