A student measures the spring constant k of a spring by loading it and measuring the extension. (According to Hooke’s law, if l is the load and e is the extension, then k=l/e.) Assume five independent measurements are made, and the measured values of k (in N/m) are 36.4, 35.4, 38.6, 36.6, and 38.0.
a. Estimate the spring constant, and find the uncertainty in the estimate.
b. Find an approximate value for the uncertainty in the average of 10 measurements.
c. Approximately how many measurements must be made to reduce the uncertainty to 0.3 N/m?
d. A second spring, similar to the first, is measured once. The measured value for k is 39.3. Approximately how much is the uncertainty in this measurement?
The average of the five measurements is 
= 37.0 and the sample standard deviation is s = 1.28841.
(a) 
(b) The uncertainty is approximately 
(c) Let n be the required number of measurements. Then 
Approximating with 
with s = 1.28841 yields s = 128841 / 
= 0.3. Solving for n yields Since n must be an integer, or 19
(d) Since the second spring is similar to the first, it is reasonable to assume that the uncertainty in a measurement of its spring constant will be similar to that of the first. Therefore the uncertainty in a single measurement is approximated with s = 1.29N/m.
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