FVCC Physics Laboratory 2011
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.
 Tape measure

3 beam balance

Weight set and hanger

Oscillating mass
 Spring
 Pendulum apparatus

Timer
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:
5.
6.
Calculate the error in g using the rules of error analysis. To do this estimate the percent error in l, and then estimate the error in 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.
7.
Calculate the accuracy in g by comparing it to 9.8m/s
2
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.
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.
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
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 estimated by calculation but that is beyond the scope of this laboratory exercise.
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.
5.
Be sure to estimate uncertainty in the measured k.
6.
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.
7.
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.
8.
Compare the k found to that measured with the simple harmonic motion data. This can be thought of as our accuracy.

FVCC Physics Laboratory 2011
7
6
5 i
L r
S
)
(
N a
4
3
2
1
0
0 y = 15.935x + 1.1123
0.1
0.2
Spring deflection (m)
0.3
0.4
Series1
Linear (Series1)

You can solve the SHM equation given above for

. Doing so gives:
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.