Experiment VII

Simple Harmonic Motion

**Introduction**

A periodic motion is one that repeats itself in successive equal intervals of time; the time required for one complete repetition of the motion being called its period. Imagine, for example, a particle moving back and forth along a straight line between two fixed points. To undergo such motion, a particle must be subject to a net force at least part of the time since the velocity of the particle is not constant.

Suppose that the net force on the particle above during its periodic motion is such that the magnitude of this force is proportional to the displacement of the particle from the midpoint of its path and is always directed toward its midpoint. That is:

F = -kx · · · · · · · (7.1)

where k is a constant and x = 0 at the midpoint. *The motion that
results from this specific net force acting on the particle is called simple
harmonic motion.*

Figure 7.1 illustrates this situation at one instant during the motion.

The points x = ± A are the end-points of the motion where A is called the amplitude.

Newton's second law, F = ma, applied to simple harmonic motion yields -kx = ma or

a = -(k/m) x. · · · · · · · (7.2)

Thus, in simple harmonic motion, both the force and the acceleration are always directed toward the midpoint. Both are zero at the midpoint and at their maximum at the end-points.

In your text it is shown that the resultant equation of motion of the mass (the equation giving the position x of the mass as a function of time) is:

x = A sin 2t/T · · · · · · · (7.3)

where t is the time elapsed starting at x = 0 when t = 0. T is the period of the motion and analysis shows that it is given by:

·
· · · · · · (7.4)

**Objective**

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

1. First you will determine if the force of the spring on the mass is of the form needed for harmonic motion, that is, 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
simple harmonic motion.

**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 below.
Place a 50 gram mass hanger on the spring and then add a 50 gram slotted
mass. Record the equilibrium position of the hanger. This will be your
*reference point* from which you will measure the spring stretch.
Note that you must start with this much mass on the spring to ensure that
all of the spring coils are separated and not squeezing each other.

Apply a series of *additional *forces to the spring ranging from
about 0.5 N to 5 N. Record these forces and the corresponding rest position
of the hanger. (Remember that W = mg so that a 50 gram mass weighs 0.49
N in your lab). Use the computer program Quattro Pro to plot the *additional*
forces added to the Hanger versus displacement of the spring from your
reference point. See Appendix C on using Quattro Pro. Click on "File",
then "Open" after you see the Quattro Pro Window. Click on the
file "SHM.wb3" and then click on "OK". Do a "Linear
Regression Fit" of W (Dependent) against displacement, x, (Independent).
Make a plot of your measurement values and the "best fit" line.

Note that your measured Y are now the added forces in Newtons and the measured x are the displacements from your reference point and should be in meters. From the "Linear Regression Fit" determine the spring constant k and the uncertainty in k.

Obtain a printed copy of your analysis and graph. What is the value
for the force constant K of your spring and what is its uncertainty? A
device that obeys the relationship F = - kx is said to obey Hooke's Law.
Does your spring appear to obey *Hooke's law* within experimental
error? (Not all stretchable materials obey Hooke's Law; a rubber band does
not nor will a spring if it is stretched so far it is permanently deformed).

2. Determine the period of oscillation for at least 5 different masses
added to your spring. It will be necessary now to *include* the mass
of the hanger as it represents mass that is in motion. Start with 150 grams
mass (including hanger) on the spring and work up to the largest mass practicable.
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. (*Remember: the count is zero when the timer
is started*).

Calculate the period T (the number of seconds per oscillation) for each
of your added masses, m_{a'} and prepare the following table:

Table 7.1

mass (m_{a}) |
Time 20 Oscillations |

1 0.150 kg | |

2 | |

3 | |

4 | |

5 | |

6 | |

7 |

Use the computer program Quattro Pro to calculate the period T and then
period squared, T^{2}. Plot T^{2} as a function of m_{a}.
Follow the steps in Appendix C - Section III and do a "Linear Regression
Fit" of T^{2} (Dependent) against m_{a} (Independent).
From the output of the linear regression, determine the slop, m and the
y-intercept, C. Also, find the percentage error of the slope.

From Equation 7.4 the theoretical relationship between T and m for a
harmonic oscillator is T^{2} = (4^{2}/k)m. If the motion
you observed was harmonic, your slope, m, should therefore equal 4^{2}/k.
Use your result for k from part 1 to see if this is the case within experimental
error. By what percent do the 2 values differ?

Although the theoretical equation T^{2} = (4^{2}/k)m
yields T = 0 when m = 0 you will note that in your graph T 0 when m_{a}
= 0. The reason for this is that the theoretical relationship is based
on a massless spring. However, since relation each part of the spring undergoes
less and less motion as the fixed end of the spring is approached, it cannot
be expected that the correction for a non-zero mass spring, can be made
by including the whole mass of the spring in T^{2} = (4^{2}/k)m.
Instead, let m_{e} be the effective mass of your spring as regards
its mass effect on T^{2}. Then m = m_{a} + m_{e}
and

T^{2} = (4^{2}/k)(m_{a} + m_{e}) = (4^{2}/k)m_{a}
+ (4^{2}/k)m_{e} · · · (7.5)

Now by using the value of T^{2} when m_{a} = 0 from
your analysis you can determine the experimental value of m_{e},
since the second term on the right (Equation (7.5)) is equal to the y-intercept..
Compare this with the total mass of your spring. About what fraction is
m_{e} of the total spring mass?

3. For harmonic motion the period does not depend on the amplitude. If time permits, perform a brief experimental check of this fact. Do this by taking the time for 20 oscillations for a fixed mass but with different amplitudes.

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© 1997 Dr. H. K. Ng.

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