Question

This procedure has been used to "weigh" astronauts in space: 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 now takes 2.54 s for one cycle.What is the mass of the astronaut?

Answer 1

Answer:

The mass of the astronaut is approximately 119.74 kg

Explanation:

Assuming this problem as a Simple Harmonic Motion of a mass-spring system, the period (T) of the oscillations for a mass (m) and spring constant (k) is:

$T=2\pi\sqrt{\frac{m}{k}}$ (1)

First, we have to calculate the spring constant using equation (1) and the data provided for the oscillations without the astronaut:

(it’s important to note that one complete vibration is the period of the movement)

$1.3=2\pi\sqrt{\frac{42.5}{k}}\Longrightarrow k=42.5(\frac{2\pi}{1.3})^{2}$

$k\approx992.8\,\frac{N}{m}$

Now with the value of k, we can use again (1) to find the mass of the astronaut (Ma) that makes the period to be 2.54 seconds

$2.54=2\pi\sqrt{\frac{42.5+M_{a}}{992.8}}\Longrightarrow M_{a}=992.8(\frac{2.54}{2\pi})^{2}-42.5$

$\mathbf{M_{a}\approx119.74\,kg}$