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Experiment 6: Hooke’s Law and Simple Harmonic Motion

Part I: Hooke’s Law

Theory:

If a spring or other sort of elastic material is subjected to a force F that displaces a distance x from it from its equilibrium position, then Hooke’s Law states that the spring or elastic material should experience a restoring force Fres that points in the direction opposing the pulling force that is linear to the displacement from equilibrium. In other words,

where k is the spring constant (sometimes called the force constant), which is measured in units of N/m. Hooke’s law is an empirical law, not a universal law. It is not true for all materials. Certain materials are nonlinear, which means that the restoring force they feel do not depend linearly on the displacement from equilibrium. They may depend on the second power of the displacement, the third power, etc., or on some other factors entirely. However, for a broad range of materials (and for a surprising number of atomic scale models) Hooke’s law is remarkably accurate.

By applying different forces to a spring, it is possible to predict the spring constant k for a given spring. Conversely, if k is known for a spring, you can predict its displacement from equilibrium for a given applied force.

Part A: Determining the spring constant k

One common way to determine k for a spring is to use the weight of a hanging mass as the known applied force. This is shown in Figure 1, along with its corresponding free body diagram

Figure 1.

If the displacement of the spring is denoted by y, then when the system at equilibrium:

Using Hooke’s law Fsp = ky, where y is the displacement from equilibrium,

Setup and Procedure:

A. Determining the spring constant

A set of springs all of the same force constant is provided. Arrange a single spring into the setup as shown in Figure 3. Attach as mass hangar to the bottom of the spring. Starting with just the mass hangar, then adding masses in small amounts (10g-20g at a time), record the weight of the mass and the corresponding displacement from equilibrium in Table 1.

Figure 3. Experimental setup. Attach coil spring to the support and attach the mass hangar to the bottom of the coil.

Spring 1

Equilibrium length: 0.05m

mass (kg)	mg (N)	y ( m ) (Displacement from Equilibrium)
0.2	1.66	0.002
0.4	3.92	0.025
0.5	4.9	0.04
0.6	5.88	0.05

Spring 2

Equilibrium length: 0.055m

mass (kg)	mg (N)	y ( m ) (Displacement from Equilibrium)
0.02	0.196	0.012
0.04	0.392	0.043
0.05	0.49	0.057
0.07	0.686	0.08

Spring 3

Equilibrium length: 0.053m

mass (kg)	mg (N)	y ( m ) (Displacement from Equilibrium)
0.02	0.196	0.33
0.04	0.392	0.05
0.05	0.49	0.087
0.07	0.686	0.125

Table 1

Part II: Finding the Period of Simple Harmonic Motion

Theory:

Once the spring constant is known, it is possible to make predictions about the behavior of the spring under a number of different circumstances. One particularly interesting and useful in which to examine a mass-spring system is when it is set into simple harmonic motion (SHM).

Simple harmonic motion (SHM) describes the oscillations of a system with a linear restoring force around an equilibrium point. It bears a great deal of similarity to the concept of uniform circular motion in terms of the period and frequency of the motion. One of the most common systems used to demonstrate SHM is the mass-spring system. In this setup, shown in Figure 4, a spring that obeys Hooke’s Law has a mass attached to one end, which is allowed to hang freely. The top of the spring is secured with a clamp. The mass on the spring is then displaced some known distance from the equilibrium position, and then released. The spring then acts upward on the mass with a force equal to the value given by Hooke’s Law.

This motion (assuming small or no losses due to friction) will be periodic. As in the case of uniform circular motion, this means that a “cycle” identically repeat over some characteristic time interval, which we call the period T.

For SHM it can be shown that the period of the motion is given by

where m is the mass and k the spring constant.

Figure 4. Experimental setup. Attach the tapered coil, hang a mass, and set it into SHM. Use small displacements. Count the time needed for 10 full cycles, and then divide that number by 10 to determine the period of oscillation.

Setup and Procedure:

1. First you must come up with an estimate for k for the spring. A tapered spring is used for this part of the experiment. It reduces friction in the system. As in Part I of the lab, add mass to the spring and record its displacement from equilibrium. Enter your values in Table 2, and determine an estimate for k by averaging together the values in the last column.

0.15m

mass (kg)	mg (N)	y ( m ) (Displacement from Equilibrium)
0.1	0.98	0.09	10.88
0.14	1.372	0.14	9.8
0.2	1.96	0.19	10.32
0.25	2.45	0.24	10.2

Table 2

kavg =

2. Now add mass to the hanger attached to the tapered spring. Displace the mass a small distance from equilibrium. (Be careful not to displace it too far, or the amplitude of motion will be too large and the mass can go flying. Record how long it takes to pass through 10 full cycles, and determine the experimental period. Calculate the theoretical period you would expect using Eq. 2. Use the value of kavg from Table 2 for your spring constant. Finally, compute the percent errors.

mass (kg)	t (sec) 10 cycles	Texp (sec) t/10	Ttheor (sec) Eq. (2)	% error
0.1	7.43	0.74	0.62	19.35
0.14	8.01	0.8	0.73	9.59
0.2	9.73	0.97	0.88	10.23
0.25	10.68	1.06	0.98	9.18

Table 3

Discussion and Analysis:

This section in your lab report should contain information that provides answers and/or explanations to the following:

1) Graph Force versus displacement for three different springs in Part I. Fit a linear curve to the data. What is the meaning of the slope? What physical characteristic of the mass-spring does it describe? Include the equations of lines with your graphs, and make sure the equations are written in terms of the correct variables. The springs you used have known spring constants of 5 N/m, 7 N/m, and 70 N/m. How do your results compare? Include percent error calculations.

2) Does the spring constant depend on how much force is applied to the spring? What does the data in your experiment suggest about it?

3) Graph T2 vs. m/k for the SHM case. To do this, carry out the following steps:

i. Take the masses in the first column of Table 3, and divide each of them by the experimentally determined value of the spring constant kavg (determined in Table 2), Record these values in the first column of Table 4 below.

ii. Square each of the values in the third column of Table 3 (the values of Texp). Enter these values in the second column of Table 4.

iii. Using the first column in Table 4 as your x-values, and the second column as your y-values, graph the data. Fit a line to this graph, and include the equation in terms of the correct variables.

4) Based on Eq. (2), what do you expect the slope to be equal to? What is the percent error between what you expect the slope to be equal to and what you observed? Include the equation of a line for this graph.

5) Identify sources of error in your experiment.