A pair of students perform a classroom investigation to explore what physical quantities affect the time it takes different oscillators to complete one cycle.
They build two separate setups:
1. A simple pendulum made by attaching a small metal bob to a light, non-stretchable string. They vary the length of the string while keeping the bob’s mass the same, then later vary the bob’s mass while holding the string length constant.
2. A horizontal spring–mass system on a nearly frictionless track. They vary the mass attached to the spring while keeping the spring the same, then later swap in springs of different stiffness while keeping the mass constant.
A summary of their averaged timing data is provided.
Pendulum Trials (small-angle displacements)
Trial
Length of string (m)
Bob mass (kg)
Measured time for 10 cycles (s)
P1
0.50
0.10
14.0
P2
1.00
0.10
20.0
P3
0.50
0.20
14.1
Spring–Mass Trials
Trial
Mass (kg)
Spring constant (N·m⁻¹)
Measured time for 10 cycles (s)
S1
0.20
50
5.6
S2
0.40
50
7.9
S3
0.20
100
4.0
Assume measurement uncertainty is negligible compared to the differences shown.
(a) Describe the relationship between the measured time per cycle of the pendulum and each of the two quantities that were varied (string length and bob mass). Support your answer with specific evidence from the data.
(b) Describe the relationship between the measured time per cycle of the spring–mass system and each of the two quantities that were varied (mass and spring constant). Support your answer with specific evidence from the data.
(c) Using your observations from parts (a) and (b), explain why changing the mass has different effects on the period of a pendulum compared with the period of a spring–mass system.
(d) For the pendulum, predict qualitatively (faster, slower, unchanged) how the frequency would change if the string length were quadrupled. For the spring–mass system, predict qualitatively how the frequency would change if the attached mass were quartered. Justify each prediction without performing numerical calculations.