Question

The formula for the up and down motion of a weight on a spring is given by s(t) = a sin (square root of) k/m (t). if the spring constant is 5, then what mass m must be used in order to produce a period of 6 seconds.

Answer 1

The period of a simple harmonic oscillator is given by

`T=2\pi\sqrt[]{(m)/(k)}`

Then, if k = 5 and the period has to be 6 seconds, we can find the mass m as:

`\begin{gathered} T=2\pi\sqrt[]{(m)/(k)} \\ (T)/(2\pi)=\sqrt[]{(m)/(k)} \\ ((T)/(2\pi))^2=(m)/(k) \\ m=k\cdot((T)/(2\pi))^2 \\ m=5\cdot((6)/(2\pi))^2 \\ m\approx5\cdot(0.955)^2 \\ m\approx5\cdot0.912 \\ m\approx4.56 \end{gathered}`

NOTE: as T is in seconds, we assume standard units for the constant k. Then, the mass is in kg.

Answer: the mass has to be approximately 4.56 kg.