Question

A pendulum has a string of length 0.58 meters supporting it. Beside it is a mass connected to a spring of spring constant 61.56 N/m. What should be the mass of the object connected to the spring so that the two oscillating systems can resonate with each other?

Answer 1

At the resonance, the frequencies (or periods) are the same.

1) The period of the simple pendulum:

$$T_1=2\pi\sqrt{l/g}.$$

2) The period of the spring pendulum:

$$T_2=2\pi\sqrt{m/k}.$$

3) Equate the periods and determine the mass:

$$\sqrt{l/g}=\sqrt{m/k},\\ m=kl/g=61.65·0.58/9.8=3.65\text{ kg}.$$