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 distance away from the equilibrium position. The minus sign indicates that the force is always pointing towards 0.  When the mass is to the right of 0, the force is to the left, and vice versa.

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

The points x = ± A are the biggest distances away from 0.   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, Fnet = 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 Fnet = 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 Fnet = -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 .

A mass oscillating on a spring is a very interesting example of energy conservation.  For some of the time the mass is moving fast, and other times the mass is stopped.  Thus, kinetic energy is converted into potential energy and then back to kinetic, over and over, right before your eyes.

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.  If it does, then you will determine k.

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.     Determine k.

Hang the spring as shown in the figure.  Make sure the fatter end of the spring is down.  Place a 50 g mass hanger on the spring and then add a 50 g slotted mass. Record the equilibrium position of the bottom 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.

Add a series of additional masses to the spring in 50 g increments. Record the added masses and the corresponding positions of the hanger (in an Excel file).  You now have a set of added masses (starting at 50 g) and positions in two columns.  Now for each new position, the mass on the spring is not moving, thus its acceleration is 0.  Thus Fnet = 0.  The forces acting on the mass are its weight and the force of the spring.  Thus W - kx = 0.  We can rewrite that equation as W = kx.  That is the equation for a straight line.

Make two new columns for the distances x from the reference point and the weights stretching the spring.  You must convert the added mass to weight (in Newtons). Do you recall how to do that?  The textbook will help if you've forgotten.

Make an Excel chart with the weight added 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 plot that best-fit line on your chart.  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 figure out and 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 a mass displaced from equilibrium should undergo Simple Harmonic Motion.

2.     Does period depend on amplitude?

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 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 ten times.  Now determine the period, the time it takes to complete one cycle.  Based on the variation in your measurements, how precisely can you report your determination of the period?

Determine the period for 5 cm and for 20 cm amplitudes.  Make at least three measurements each.  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 your measurements?

3.     How does period depend on mass?

Remove the ruler from the hanger.  Determine the period of oscillation for at least 8 different masses hanging from your spring. Start with 150 g (including hanger) on the spring and work up to the largest mass you can (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. Plot T vs m.  Do the data follow a straight line?

The equation suggests that plotting T vs should be a straight line of slope .   If so then a plot of T vs m should not be a straight line.  Make a new column for .  Plot T vs .   Since the spring is the same as you used in part 1, the spring constant k should be the same value (check units) as you got in part 1.  Use the value of k you determined in part 1 to calculate the slope you would predict for the T vs  plot.  Put that slope in an Excel cell.  Make another cell for the Y-intercept.  Make a new column for your proposed fit for T.   This proposed fit should be of the form .  By looking at your plotted data, try to estimate the Y-intercept, and put that value into its cell.  Do you get a good fit?  Vary the value of Y-intercept to get a good fit.

4.     Pendulum experiment.

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 string or the mass hung from the string. Then continue the experiment to determine how the period depends on the more important parameter.