Simple Harmonic Motion
in a metal plate
Abstract:
Measure the period of various masses oscillating on a metal plate to determine
its force constant.
Introduction:
Assuming the plate is massless, it would have a restoring force in accordance to
Hooke’s law
F = -kx
Thus by finding T2 in relation to masses, one need simply to produce a linear
curve fit on the points created to find force constant k.
Diagram:
Procedure:
1. Place a mass at the free end of the metal plate
2. Set the photogate timer in such a fashion so as to be barely above the point at
which it detects the plate (the goal is to catch the plate at the top of every
oscillation, thus having every spike represent one period)
3. Oscillate the plate
4. In the data recorded, use an arbitrary beginning point and find the associated
point exactly ten periods away. Subtract to find time for ten periods
5. Repeat (more often for more accuracy), and take the average
Analysis:
Using the initial trials, averages were gathered for two trials per mass:
Mass
Initial Observed
Time
Initial Data
Final Observed
Time
Time for Ten
Oscillations
Using the averages, a final table of data was compiled:
Mass (kg)
Final Data
Average Time for
10 Vibrations (s)
Period
T (s)
T Squared (s)
Average
Using the final values for T Squared and Mass, a graph of T Squared vs Mass was
created and curve-fitted:
The line is very much linear, and yields the equation
Using the basic equations for Simple Harmonic Motion,
A = -w2x
A = -(k/m)x
One yields
w = sqrt(k/m)
T = 2pi*sqrt(m/k)
By squaring both sides,
T2 = (4pi2/k) M
Thus, using both equations,
T2 = (4pi2/k) M
T2 = _______M
One can equate the two slopes,
(4pi2/k) = _________
k = (4pi2/
)=
Conclusion:
The force constant was found to be ___________
Sources of Error:
The prime source of error would be the fact that the plate itself is not massless,
which is an assumption in this lab. Thus is most likely the source of the y-intercept in
the curve fit. Also, data collection could be more reliable by taking more consistent start
and end points for time.