A student investigates how mechanical energy in a spring–mass system varies with amplitude
PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics
Problem
A student investigates how mechanical energy in a spring–mass system varies with amplitude. Using the same cart and spring throughout, the student measures the total mechanical energy for several amplitudes and records the data below.
Amplitude A (m)
0.03
0.05
0.07
0.09
Total Energy E (J)
0.11
0.30
0.74
1.22
Assume that energy is measured just after release from rest at each amplitude and that all motion is frictionless.
(a) Demonstrate quantitatively that the data are consistent with a proportional relationship between E and A². Clearly show the calculation you use to support your claim.
(b) Treating the relationship as linear in the variables E and A², determine the spring constant k for the system. Outline any plotting or algebraic steps you employ.
(c) Using your value of k, predict the amplitude required for the system to possess 2.0J of total mechanical energy when released from rest.
(d) Suppose the experiment is repeated with a new spring whose constant is exactly twice the value found in part (b). Describe how the slope of an E versus A² graph would change, and explain your reasoning.