Question

Space-faring astronauts cannot use standard weight scales (since they are constantly in free fall) so instead they determine their mass by measuring the period of oscillation when sitting in a chair connected to a spring. Suppose a chair is connected to a spring with a spring constant of 600 N/m. If the empty chair oscillates with a period of 0.9s, what is the mass of an astronaut who oscillates with a period of 2.0 s while sitting in the chair

Answer 1

Answer:

ma = 48.48kg

Explanation:

To find the mass of the astronaut, you first calculate the mass of the chair by using the information about the period of oscillation of the empty chair and the spring constant. You use the following formula:

$T=2\pi\sqrt{\frac{m_c}{k}}$     (1)

mc: mass of the chair

k: spring constant = 600N/m

T: period of oscillation of the chair = 0.9s

You solve the equation (1) for mc, and then you replace the values of the other parameters:

$m_c=\frac{T^2k}{4\pi^2}=\frac{(0.9s)^2(600N/m)}{4\pi^2}=12.31kg$    (2)

Next, you calculate the mass of the chair and astronaut by using the information about the period of the chair when the astronaut is sitting on the chair:

T': period of chair when the astronaut is sitting = 2.0s

M: mass of the astronaut plus mass of the chair = ?

$T'=2\pi\sqrt{\frac{M}{k}}\\\\M=\frac{T'^2k}{4\pi^2}=\frac{(2.0s)^2(600N/m)}{4\pi^2}\\\\M=60.79kg$ (3)

Finally, the mass of the astronaut is the difference between M and mc (results from (2) and (3)) :

$m_a=M-m_c=60.79kg-12.31kg=48.48kg$

The mass of the astronaut is 48.48 kg