KVPY Sample Paper KVPY Stream-SX Model Paper-22

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) \[0.92{{s}^{-1}}\]

Correct Answer: A

Solution :

\[f\propto {{m}^{a}}{{k}^{b}}\]

\[{{T}^{-1}}={{M}^{a}}{{M}^{b}}{{T}^{-2b}}\]

\[a+b=0;-2b=-1\Rightarrow b=1/2,a=-1/2\]

\[f\propto \sqrt{\frac{k}{m}};\frac{0.50}{f}=\frac{\sqrt{85}}{\sqrt{72}}\Rightarrow f=\frac{1}{2}\sqrt{\frac{72}{85}}=0.46{{s}^{-1}}\]