FVCC Physics Laboratory 2011
Simple Harmonic Motion
Objective:
By measuring the period of simple harmonic motion, we will determine the acceleration
due to gravity with a pendulum. Then with an oscillating mass system, we will measure
oscillation period and use it to determine
the spring constant K of a mass-spring
system. This K can be checked by
stretching the spring with various weights.
Background: See Physics Text.
Equipment:
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Tape measure
3 beam balance
Weight set and hanger
Oscillating mass
Spring
Pendulum apparatus
Timer
Procedure:
A) The Pendulum: Measure g
This part of the lab will measure the acceleration due to gravity using a simple pendulum.
1. Set-up a long length pendulum ( 80 cm range), and measure the exact length of the
string. Measure the pendulum length from the point where the string is attached to the
vertical frame to the center of the mass at the end.
2. Being sure to use small amplitudes, start the pendulum swinging.
3. Find the period of oscillation. The best way to do this is to time the pendulum while it
goes through many complete oscillations. This will reduce the error in the
FVCC Physics Laboratory 2011
measurement. Try counting 30-40 oscillations and time how long it takes to complete
these ‘N’ cycles. Period = Total time/N.
4. Calculate the value for g from:
2𝜋 2
5. 𝑔 = ( 𝑇 ) 𝑙
6. Calculate the error in g using the rules of error analysis. To do this estimate the percent
error in l,
∆𝑙
𝑙
and then estimate the error in T,
∆𝑇
𝑇
. Note that the error in T is best
estimated by assuming the error is in loosing count. So if you counted 50 periods, it is
reasonable to assume that you lost count by one. This makes the error in period 1/50 or
.02. Your error will be different if you use a different count. There is also a little bit of
error in the timer but it is safe to assume the error here is negligible. The uncertainty in
g is then given by equation 1. This will be the uncertainty in g.
∆𝑔
2∆𝑇 2
∆𝑙 2
√
= (
) +( )
𝑔
𝑇
𝑙
7. Calculate the accuracy in g by comparing it to 9.8m/s2
B) The mass-spring
In this part of the lab, we will use simple harmonic motion to measure the spring constant K
and then compare our result with the K determined by measuring spring deflection directly.
For a mass spring system we have the period of oscillation given by equation 2.
𝑇 = 2𝜋√
𝑚
𝐾
This equation is close to correct but not exact. The problem is that m is considered all of the
mass that is suspended at the end of the spring. But the spring is not mass-less, so the weight
of the spring contributes a small portion of its weight to the mass we need to use in equation 2.
As a result, a more accurate equation is given in equation 3.
(𝑚 + 𝜀𝑚𝑠 )
𝑇 = 2𝜋√
𝐾
Here  is a fraction ranging from 0 to 1 which will account for that portion of the springs mass
we should include in the calculation of the period. It is experimentally determined but we can
use =0.5.
FVCC Physics Laboratory 2011
2𝜋 2
𝐾 = (𝑚 + 𝜀𝑚𝑠 ) ( )
𝑇
Note: Large springs with weak K are more effected by this. Springs that are light yet strong will
be less effected by the error introduced by the empirical value for . This value can be estimate
by calculation but that is beyond the scope of this lab exercise.
Measure K with Simple Harmonic Motion
1. Measure the mass to hang from the spring (m), and the mass of the spring itself m s.
2. Hang a mass on the spring and start it in SHM. Note that the small amplitude restriction
in the pendulum is not a restriction here.
3. Find the period of oscillation in the same way you found the period in the pendulum.
Specifically, time the system while it makes N oscillations. Then the period is T= time/N.
4. Calculate the value of K using equation 3. This is the K estimated from SHM.
Measure K by loading spring
5. Now find K by suspending weight from the spring. Measure the spring constant by
hanging a series of weights from the spring and measuring the subsequent elongation of
the spring. Note, since you are measuring spring elongation, you need to establish an
initial position for the end of the spring and then all other measurements are relative to
that point. Be sure that the spring has enough weight on it initially to stretch it a little.
6. Plot the force on the spring as a function of the elongation in the spring. Find the slope
of the plot and use this for your measured value of the spring constant. This can also be
done by linear regression. Figure 2 shows an example plot where the slope is K, the
spring constant. The initial weight is simply the y intercept on the Force vs. Distance
plot.
FVCC Physics Laboratory 2011
7
y = 15.935x + 1.1123
6
5
) 4
N
(
d
a 3
o
L
g 2
in
r
p
S 1
Series1
Linear (Series1)
0
0
0.1
0.2
0.3
0.4
Spring deflection (m)
Figure 2. Example plot for determining spring constant
C) OPTIONAL: Finding the  value for the spring
You can solve the SHM equation given above for . Doing so gives:
𝑇 2
𝐾 (2𝜋) − 𝑚
𝜀=
𝑚𝑠
Measuring the various quantities in the equation would then allow you to determine an exact
value for . Springs measured in the lab range from .32 to .6. Further, one spring in the lab
which has a conical shape verses the cylindrical shape of most springs, gives .44 when oriented
one way and .33 when reversed. The measurement is reasonably straight forward with the
simple caution that light springs with strong K values produce a measurement that is very
delicate. This is a consequence of measuring a very small number. It is advisable to measure T
very carefully so that you have enough significant figures to make a good estimate of a very
small effect.