Experiment VII - Simple Harmonic Motion

Simple Harmonic Motion

Introduction

Simple harmonic motion (SHM) is the back and forth motion of objects around a point. Swings, trees swaying in the breeze, pendula, a boat bobbing on the ocean, and the swinging of your relaxed arm as you walk are all examples of SHM.

For SHM to occur there must be a force that pushes the object back to a preferred (equilibrium) position. When that force is proportional to the distance away from equilibrium, then SHM can happen. Mathematically we write this as

F = -kx

where k is a constant and x is the displacement distance away from the equilibrium position. The minus sign indicates that the direction of the force is back toward the equilibrium, opposite the displacement.

Figure 1 illustrates a mass at one instant during its back and forth motion between A and -A.

The points x = ± A are the end-points of the motion. A is called the amplitude. The back and forth motion is also called oscillation of the mass.

A mass on the end of a spring is an excellent example of a system that will exhibit SHM. Newton's second law, F_net = ma, applies to a mass hanging from a spring. If we hang a mass from a spring and let it come to rest, then two forces are acting but there is no acceleration F_net = mg - kx = 0, and the added weight will equal the spring constant times the distance the spring moved. This relationship is used for the first experiment.

When we pull the spring away from equilibrium and let it go, then the only force (away from equilibrium) is the restoring force F_net = -kx and this will equal ma. This is used for the second experiment, when the motion repeats itself. The time for one repeat cycle is called the period. If we call the period of the motion T, it can be shown that

Objective

In this experiment you will study whether the vertical motion of a mass dangling from a spring is a good approximation to SHM by making two sets of measurements.

1. First you will measure the distance the spring stretches when different weights are added to see if F = - kx.

2. Secondly, you will measure the period of the motion for several different masses attached to the spring to see if the period varies in accord with the result for SHM.

Apparatus

Electrical timer, set of slotted weights and hanger, spring, meter stick, table clamp, rod and support.

Procedure

1. Hang the spring with the large end down as shown in the figure. Place a 50 g mass hanger on the spring and then add a 50 g slotted mass. Record the equilibrium position of the hanger. This will be your reference point from which you will measure the spring stretch. You must start with this 100 g on the spring to ensure that all of the spring coils are separated and not squeezing each other.

Apply a series of additional masses to the spring in 50 g increments. Record these masses and the corresponding positions of the hanger (in an Excel file would make sense). You now have a set of positions and masses in two columns.

Find the distances x from the reference point and the forces F that you needed to add for each of those readings. To find the force added, you must convert mass to weight (in Newtons). Do you recall how to do that? The textbook will help if you've forgotten.

Make an Excel plot with the force along the Y axis and the distance away from the reference point along the X axis. Find the slope and intercept of a best fit straight line through your data using the LINEST command and calculate and plot that best fit line. The slope of the line is the magnitude of the force constant of the spring. What is k and what is a reasonable estimate for your uncertainty in k? Don't forget to write the units of k.

If your data did not form a fairly good straight line, you should try again! If they did form a straight line, then you have demonstrated that the spring stretches in proportion to the force applied. Thus, a mass displaced from equilibrium should undergo SHM.

2. Now hang about 400 g from the spring. Push the mass up 10 cm and release it. What happens? Pull the mass down 10 cm and release it. What happens?

While the mass is oscillating with about a 10 cm amplitude, measure how long it takes to complete 20 cycles. (The easiest way to do this is to start counting 0 at the top (or bottom) of a cycle, and 1 at the next top (or bottom), continuing to count up to 20. Repeat the measurement, to make sure it seems correct. Now determine the period, the time it takes to complete one cycle.

Repeat this procedure for 5 cm and 20 cm amplitudes. Are your data consistent with the statement, "For simple harmonic motion the period should not depend on the amplitude"? Be careful in answering this. Of course your measurements of period are going to vary. Do they vary by more than the uncertainty of the measurement?

3. Remove the ruler from the hanger. Determine the period of oscillation for at least 7 different masses hanging from your spring. Start with 150 g (including hanger) on the spring and work up to the largest mass practicable (at least 800 g). Do not give the oscillation such a large amplitude that the coils of the spring hit together at the top of the motion. Time at least 20 oscillations for each load and repeat. (Count zero when the timer is started). Record these data in Excel.

For each mass, calculate the period T, the time for one oscillation, and also calculate the period squared, T². Plot T as a function of m. Do the data follow a straight line? Plot T² as a function of m. Do the data follow a straight line? Plot both cases using Excel and try for linear fits using LINEST to find the slope and intercept. Note: it is impossible for both T and T² vs m to be straight lines. If they both appear to be straight, then either your range in mass is not large enough or you have done something incorrectly.

The equation suggests that plotting T vs , or T²vs m should look like a straight line. Are your data consistent with the theory? When you plot T² vs m you had to square the equation. Thus the slope = . Solve this equation for k. It should be the same value (check units) as you got in part 1. Is it the same(within experimental uncertainty)?

4. A pendulum consists of a mass hung from the end of a string, which is attached to a fixed support. When released from any position, but plumb, the mass will go into oscillation.

Please create an experiment (or several) to determine what affects the period more: the length of the or the mass hung from the string.