Theory
A weighted spring is hung vertically, subjected to a displacing force, and released. The motion of its weighted end, plotted as a function of time, resembles that of a simple linear oscillator:
h = A cos(B t + C) + D
where h is the vertical displacement from a fixed reference
A is the displacement amplitude
B the angular frequency [radians per second]
C is the phase angle [radians]
D is the relaxed position distance from reference
The angular frequency B is entertaining. Its negative square is the ratio of the spring’s endpoint acceleration a to its endpoint displacement x (where both are functions of time):
a = - B2 h
Multiplying both sides by the effective mass m of the spring/weight system,
m a = - m B2 h
Hooke’s Law relates the vertical spring’s restoring force F at any instant to its vertical displacement from relaxed position, by means of the spring constant k :
F = - k h
Evidently k is a constant value that relates a spring system’s mass to its characteristic angular frequency of oscillation:
k = m B2
Practical concerns
We have been trained to recognize k as having the unit of force divided by length (newtons per meter in S.I.), which simplifies to mass divided by the square of time ( kg / s2 ). Since the unit of B is radians per second (s -1 ), then the unit of (m B2) is ( kg / s2 ); or, if cgs units are used, ( g / s2 ).
Angles and waves
The frequency of a rotating angle is expressed in radians per second (s -1). The y-component of a unit-circle position vector obeys the time function
y = sin t ,
so the angular frequency of a sine wave ( B s -1) equals the frequency of the rotating angle from which it was generated. In one cycle the rotating angle sweeps out 2 pi radians, so its corresponding sine wave has periodic frequency B / (2 pi ) cycles per second, and a cyclic period
T = 2 pi / B ,
where the unit of T is seconds (s) . From the above discussion, we saw the relation between a spring’s effective mass and the angular frequency of its oscillation:
B2 = k / m
Combining these two equations, we get the period of the sine wave generated by an oscillating spring:
T = 2 pi (m / k) ½
Apparently the period is determined only by the spring’s effective mass and its “stiffness” ( k ). If k for a spring is known, the period of the spring’s oscillation can be calculated from its effective mass.
Finding the value of k
From the above equation we get
k = 4 pi 2 m / T2
With mass in grams and T in seconds, the unit of k ( g / s2 ) agrees with the above discussion. But Hooke’s Law shows
k = - F / h
and the two expressions are equivalent, as we have seen. The factor k that relates a spring’s effective mass to its oscillation period, is the same k that relates a spring’s restoring force to its endpoint displacement from rest position.
We can determine the value of k for a spring by hanging various weights on it and measuring its endpoint displacement. Plotting weight as a function of displacement, the slope of the best-fit line gives the value of k . Using linear regression and a t-test, we can construct a confidence interval for our experimentally-derived value of k :
[Confidence Interval] = k ± ( t s / {Sx½} )
where k is the slope of the regression (best-fit) line
t is Student’s test statistic
s is the standard deviation of the regression slope
n is the number of samples
Sx is a certain calculated parameter[1]
(In determining k for our spring, we used the t-value for 90% confidence and 7 degrees of freedom.)
Error Analysis
In the determination of k , each of our length measurements was subject to error of no less than ± 0.1 cm. These measurement errors correspond to relative errors ranging from 1% (for larger displacements) to 5% (for smaller displacements). Since we determined k by a best-fit slope, a second source of error was in deviation of the data from linearity. In propagating measurement error through the regression, the confidence intervals for k varied like this:
|
Measurement |
90% Confidence Interval for k |
Relative error in C.I. |
|
+ 0.1 cm |
(9.8 ± 0.1 ) x 103 g / s2 |
± 1 % |
|
0 cm |
(9.82 ± 0.08) x 103 g / s2 |
± 0.8 % |
|
- 0.1 cm |
(9.83 ± 0.06) x 103 g / s2 |
± 0.6 % |
Interestingly, the effect of measurement error on the derived value of k was nearly an order of magnitude smaller than the worst-case variation in k due to nonlinearity. The nonlinearity reflects a more conservative estimate of error in the derived value of k .
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We have described the spring/mass system’s behavior in three ways:
· A sine wave
· An acceleration proportional to displacement
· A force proportional to displacement