• # question_answer The space shuttle astronauts use a 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 $\operatorname{kg}/{{s}^{2}}$and the chair's frequency $f$ is 0.50 ${{\operatorname{s}}^{-1}}$ for a 62 kg astronaut, what is the chair's frequency for a 75kg astronaut? The chair itself has a mass of 10.0kg. A) $0.46{{s}^{-1}}$ B) $0.12{{s}^{-1}}$ C) $0.78{{s}^{-1}}$ D) $0.92{{s}^{-1}}$

 $f\propto {{m}^{a}}{{k}^{b}}$ ${{T}^{-1}}={{M}^{a}}{{M}^{b}}{{T}^{-2b}}$ $a+b=0;-2b=-1\Rightarrow b=1/2,a=-1/2$ $f\propto \sqrt{\frac{k}{m}};\frac{0.50}{f}=\frac{\sqrt{85}}{\sqrt{72}}\Rightarrow f=\frac{1}{2}\sqrt{\frac{72}{85}}=0.46{{s}^{-1}}$