The Space Shuttle astronauts use a massing chair to measure their mass. The chair is attached to a spring and is free to oscillate back and forth. The frequency of the oscillation is measured and that is used to calculate the total mass \(m\) attached to the spring. If the spring constant of the spring \(k\) is measured in \(\mathrm{kg} / \mathrm{s}^{2}\) and the chair's frequency \(f\) is \(0.50 \mathrm{s}^{-1}\) for a \(62-\mathrm{kg}\) astronaut, what is the chair's frequency for a 75 -kg astronaut? The chair itself has a mass of 10.0 kg. [Hint: Use dimensional analysis to find out how \(f\) depends on $m \text { and } k .]$

The mass of the 62 kg astronaut (\(m_{62}\)) = 62 kg The mass of the 75 kg astronaut (\(m_{75}\)) = 75 kg The frequency for the 62 kg astronaut (\(f_{62}\)) = 0.50 s\(^{-1}\) The mass of the chair (\(m_c\)) = 10.0 kg Spring constant (\(k\)) = unknown Frequency for the 75 kg astronaut (\(f_{75}\)) = unknown

First, we need to add the mass of the chair to the mass of each astronaut to find the total mass attached to the spring. Total mass with 62 kg astronaut (\(M_{62}\)) = \(m_{62} + m_c\) = 62 kg + 10 kg = 72 kg Total mass with 75 kg astronaut (\(M_{75}\)) = \(m_{75} + m_c\) = 75 kg + 10 kg = 85 kg

We are given the frequency for the 62 kg astronaut, and we can use it to find the spring constant k. Hooke's law defines the spring constant as the force divided by the displacement: \(F = kx\) And the force acting on a spring-mass system is given by: \(F = m \omega^2 x\) Here, \(\omega\) is the angular frequency, and it is related to the given frequency \(f\) by \(\omega = 2 \pi f\). Plugging this into the previous equation, we get: \(kx = m(2 \pi f)^2 x\) Notice that the displacement x cancels out: \(k = m(2 \pi f)^2\) Now, we can use the values for a 62 kg astronaut to find k: \(k = M_{62}(2 \pi f_{62})^2\) \(k = 72 \text{ kg} (2 \pi (0.50 \ s^{-1}))^2\) \(k \approx 142.2 \ \frac{\text{kg}}{\text{s}^2}\)

Now we know the spring constant k, and we can use it to determine how the frequency depends on the total mass \(m\). Rearrange the expression for k to find f: \(f^2 = \frac{k}{(2 \pi)^2m}\) Notice that f depends on the total mass m and k, and it is inversely proportional to m.

Finally, we have all the information required to find the frequency for a 75 kg astronaut. We can use dimensional analysis and the relation between f and m to find the frequency: \(f_{75}^2 = \frac{k}{(2 \pi)^2 M_{75}}\) \(f_{75}^2 = \frac{142.2 \ \frac{\text{kg}}{\text{s}^2}}{(2 \pi)^2 (85 \ \text{kg})}\) \(f_{75} = \sqrt{\frac{142.2 \ \frac{\text{kg}}{\text{s}^2}}{(2 \pi)^2 (85 \ \text{kg})}}\) \(f_{75} \approx 0.453 \ \text{s}^{-1}\) The chair's frequency for a 75 kg astronaut is approximately 0.453 s\(^{-1}\).