Experiment 13
Simple Harmonic Motion
Introduction
The purpose for this experiment was to measure spring’s constant of a spring. In part 1, extension of the spring when a mass is hanged on it was measured. Six measurements for the extension were obtained for each mass used. All the measurements were used to determine the spring’s constant by plotting a graph of weight against extension. In part 2, six measurements of period were taken using different masses (same masses used in part 1). The measurements were then used to plot a graph of square of periods against mass. The equation of the graph was then used to calculate spring’s constant. The spring’s constant measured in part 1 was then compared to that measured in part 2.
Data
The following data were obtained from the experiment:
Data analysis
Value of spring’s constant (k2);
Using the equation of T2 against mass, we get;
y=1.3425x+0.0533
T2=4π2km+4π2kmeff , in which 4π2k corresponds to the coefficient of x; 1.3425
Therefore, k=4π21.3425
k=4*3.14159321.3425 = 4*9.8696041.3425 =39.4784161.3425 =38.48
Therefore, k2 = 38.48
Error in k2 (σk2) = slope of graph of T2 against mass/square of the slope
σk2= 1.3425/ 1.80230625
σk2 = 0.7
In order to compare k2 and k1, we do t-test. In this case, we should find the calculated t-value and compare it to the tabulated t-value;
t=k2-k1σd Where σd is the error in the difference between k1 and k2
σd=σk2^2-σk1^2
σd=√( 0.72+0.52 = √ 0.74
σd = 0.9
t=38.48-32.170.9
t=6.310.9
t=7.01
The calculated t-value in this case is 7.01
Next, we look for the tabulated t-value at 99% confidence level. In this case, we must know the degree of freedom.
The tabulated value as determined from the table is = 3.17
The calculated T-value (7.01) is greater than the tabulated T-value (3.17). Therefore, there is significant difference between the two values of the spring’s constant. K2 is greater than k1.
The graph of T2against mass is as shown below:
Discussion
Errors in the experiment were caused by the inaccuracy of the stopwatch used in recording periods and the tendency of the spring to fail to retain the original length after a mass is removed. The later source is a random error while the former is a systematic error. Inaccurate stopwatch leads to measurement of periods that deviate from the actual value by a given fixed value. On the other hand the spring’s loss of elasticity causes recording of inaccurate values for extension.
Questions
Question 4 in part 1
The spring constant is the slope of the graph of weight against extension and its value has been obtained to be 32.17N/m, the spring’s constant is calculated as follows
mg = -kx
Using trial 2 in part 1;
-k=0.59290.023
k= 25.77N/m
Question 7 part 2
The y-intercept is not zero.
Conclusion
The experiment was successful. The difference between theoretical value and the calculated value was found to be 6.31. On the other hand, the error in the difference was found to be 0.9. Since the difference is less than the error, the two values of spring’s constant do not agree within experimental error. This implies that the experiment was not very accurate.
