A 42.5-kg chair is attached to a spring and allowed to oscillate. When it is empty, the chair takes 1.30 s to make one complete vibration. But with an astronaut sitting in it, with her feet off the floor, the chair takes 2.54 s for one cycle. What is the mass of the astronaut?

Period of oscillation can be defined as the time taken by the object to complete one full cycle of its periodic motion.

Mathematically it is given by,

$T=2\pi \sqrt{\frac{m}{k}}$

Here, $T$is the period of oscillation in seconds

$m$is the mass of object in kilograms

$k$is the spring constant of spring

The expression for period of oscillation can be rewritten as

$k=\frac{4{\pi }^{2}m}{T^2}$

Substitute values in above expression to get the value of spring constant

$$\begin{gathered} k=\frac{4{\pi }^2\left(42.5kg\right)}{{\left(1.30s\right)}^2} \\ =992.80N/m \end{gathered}$$

The expression for period of oscillation can be rewritten for mass as

$M=\frac{{\left(T\right)}^{2}k}{4{\pi }^2}$

Substitute values in above expression to get the mass of the object when it takes 2.54 s for one cycle

localid="1668081835830" $$\begin{gathered} M=\frac{{\left(2.54s\right)}^2\left(992.80N/m\right)}{4{\pi }^2} \\ =162.24kg \end{gathered}$$

Mass of the astronaut can be calculated by,

$$\begin{gathered} m_{austronant}=162.24kg-42.5kg \\ =119.74kg \\ \approx 120kg \end{gathered}$$

Therefore, mass of the astronaut is approximately $120kg$.