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  • Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a si

Assume that the earth is a solid sphere of uniform density and a tunnel is dug along its diameter throughout the earth. It is found that when a particle is released in this tunnel, it executes a simple harmonic motion. The mass of the particle is 100 g. The time period of the motion of the particle will be (approximately) 
\left(\text { Take } \mathrm{g}=10 \mathrm{~m} \mathrm{~s}^{-2}, \text { radius of earth }=6400 \mathrm{~km}\right)

Option: 1

12 hours


Option: 2

1 hour 40 minutes


Option: 3

24 hours


Option: 4

1 hour 24 minutes


Answers (1)

best_answer

Inside earth, force is given byF = -\frac{G M_e m x}{R_e^3}
And g_0 (on surface of earth) = \frac{G M_e}{R_e^2}

\begin{aligned} & \therefore F=-\frac{g_0 m}{R_e} x \\ & \Rightarrow a=-\frac{g_0}{R_e} x \\ & \omega=\sqrt{\frac{g_0}{R_e}} \\ & \Rightarrow T=2 \pi \sqrt{\frac{R_e}{g_0}}=2 \pi \sqrt{\frac{6400 \times 10^3}{10}}=2 \times 3.13 \times 8 \times 10^2 \mathrm{sec}=5024 \mathrm{sec}=1.4 \mathrm{hr} \\ & T=1.4 \mathrm{hr}=1 \mathrm{hr} 24 \text { minutes } \end{aligned}

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sudhir.kumar

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