Physics for the Life Sciences

Experiment 15

Simple Harmonic Motion

Purpose

To use a spring to test Hooke’s law and to study simple harmonic motion.

Apparatus

Vibrating spring apparatus consisting of stand, spring, weight hanger, set of slotted weights (20 to 200 g), balance, stopwatch

Theory

When an elastic body is distorted by some externally applied force, a restoring force is produced in the body which is directly proportional to the amount of distortion caused. The relation between the restoring force F and the distortion x is F = kx. This is known as Hooke’s law and the constant k is called the force constant of the spring.
The period of vibration of a mass hanging from a vertical spring when set in simple harmonic motion is

where
T is the period of oscillation in seconds,
m is the mass of the vibrating system in kg,
k is the force constant of the spring in units of N/m.

Procedure

1. Remove the spring from the weight hanger and determine its mass in grams by weighing it on the balance to the nearest 0.1 g.
2. Weigh the weight hanger and record its mass in grams.
3. Reattach the spring to the weight hanger, suspend it from the support and adjust the scale position so that the pointer on the weight hanger reads zero on the scale.
4. Place 20 g on the weight hanger and record the position of the pointer on the scale.
5. Repeat procedure (4) with loads of 40 to 200 g in steps of 20 g in each case recording the position of the pointer on the scale.
6. Remove the 100 g from the weight hanger so that it now has only 100 g on it. Displace the weight hanger downwards about 1 cm from its equilibrium position and set it in vertical oscillation. Using the stopwatch, measure the time taken for 25 complete oscillations and record it. Remember to count zero when you let go of the mass.
7. Repeat procedure (6) with the same mass but now displace it 2 cm. Again measure the time for 25 oscillations and record its value.
8. Now add an additional 50 g to the weight hanger, displace it 1 cm and time 25 complete oscillations.
Calculations

1. For each set of data calculate the total mass suspended from the spring by adding the mass of the hanger to the added mass and calculate the stretching force in N from grams x 0.0098.
2. Plot a graph of the values of the total stretching force in N (on the y-axis) versus the corresponding value of the scale reading in cm (on the x-axis). The slope of the graph gives the force constant of the spring k in N/cm. Convert this to N/m by multiplying by 100 and enter the value in the table.
3. From the data in procedures (6) through (8) calculate the experimental values of the period (the time for one oscillation) and record the values in the table.
4. Use the equation to calculate the values of the theoretical period. (m is the mass of the vibrating system which is the weight hanger, plus the added mass, plus 1/3 the mass of the spring) and k is the slope of the graph in N/m.
5. Calculate the percent error between the experimental and theoretical values of the period using the relation
Percent error =

Data
Mass of spring = ________ g    Mass of weight hanger = ________ g

 Mass added to weight hanger in g Total mass suspended from spring in g Total stretching force in N (kg x 9.81) Scale reading in cm 0 20 40 60 80 100 120 140 160 180 200

Force constant of spring = ________ N/m

 Mass added to weight hanger in g Effective mass of vibrating system in g Amplitude of vibration in cm Time for 25 vibrations in seconds Experimental value of period in seconds Theoretical value of period in seconds Percent error 100 1 100 2 150 1

Questions

1. What does your graph show about the dependence of the elongation of the spring on the applied force? Does this verify Hooke’s Law?

2. At what point does the vibrating mass have
(a) greatest acceleration
(b) least acceleration
(c) greatest velocity
(d) least velocity
(e) greatest kinetic energy
(f) least kinetic energy ?

3. Is the motion of the piston in a gasoline or steam engine simple harmonic motion? Explain your answer by comparison to the vibration of a mass on a spring.

4. What can you deduce about the effect of the amplitude of vibration on the period of oscillation from the measurements that you took in this experiment? Is this what you expected? If not - why not?

5. Calculate the value of the mass in grams that you would need for your vibrating system to have a period of oscillation of exactly one second.

6. If the frequency of oscillation of a 405 g mass on a certain spring is 2.5 Hz, what is the value of the spring constant in N/cm?