Question

A single spring with a 0.75 kg mass on it is stretched out and then released. Then the velocity of the mass is measured as the spring passes through its equilibrium position. This experiment is repeated 18 times; pulling the spring to a different distance each time. The amplitudes and speeds are listed below. Calculate the average spring constant.  

Amplitude (m.)	Speed at Equilibrium (m/s)
0.05	0.13
0.10	0.26
0.15	0.38
0.20	0.52
0.25	0.63
0.30	0.76
0.35	0.91
0.40	1.05
0.45	1.20
0.50	1.28
0.55	1.40
0.60	1.54
0.65	1.68
0.70	1.85
0.75	1.90
0.80	2.12
0.85	2.19
0.90	2.32

Answer 1

Mass of block = m = 0.75 kg

Spring constant = k

Angular frequency of oscillation =

Amplitude of oscillation = A

Speed of the mass at equilibrium position = V = A

= V/A

k = m²

Amplitude A (m)	Speed at equilibrium V (m/s)	Angular frequency = V/A (s-1)	Spring Constant k = m2 (N/m)
0.05	0.13	2.6	5.07
0.10	0.26	2.6	5.07
0.15	0.38	2.533	4.813
0.20	0.52	2.6	5.07
0.25	0.63	2.52	4.763
0.30	0.76	2.533	4.813
0.35	0.91	2.6	5.07
0.40	1.05	2.625	5.168
0.45	1.20	2.666	5.333
0.50	1.28	2.56	4.915
0.55	1.40	2.545	4.858
0.60	1.54	2.566	4.94
0.65	1.68	2.584	5.01
0.70	1.85	2.642	5.235
0.75	1.90	2.533	4.813
0.80	2.12	2.65	5.267
0.85	2.19	2.576	4.977
0.90	2.32	2.577	4.98

Average spring constant is the average of all the calculated 'k' values.

Average spring constant = 5.01 N/m