91Ó°ÊÓ

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,

$T=2\mathrm{Ï€}\sqrt{\frac{m}{k}}$

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

$k=\frac{4{\mathrm{Ï€}}^{2}m}{T^2}$

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

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

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

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

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

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

Mass of the astronaut can be calculated by,

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

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