**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, *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

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

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 *F _{net}
= 0.* The forces acting on the mass
are its weight and the force of the spring.
Thus

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

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. (C*ount 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.

This page last updated on 27 November
2001.

© 2000,2001 David H. Van Winkle.

All Rights Reserved.