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Question
A tunnel is dug along a diameter of the earth. Find the force on a particle of mass m placed in the tunnel at a distance x from the centre.
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Solution
Mass of the Earth,
\[M = \left( \frac{4}{3} \right)\pi R^3 \rho\] ...(i)
Consider an imaginary sphere of radius x with centre O as shown in the figure below :
\[\text { Mass of the imaginary sphere }, M' = \left( \frac{4}{3} \right)\pi x^3 \rho . . . (ii)\]
\[\text { From (i) and (ii), we have : }\]
\[\frac{M'}{M} = \frac{x^3}{R^3}\]
∴ Gravitational force on the particle of mass m is given by F \[= \frac{GMm}{x^2}\]
\[\Rightarrow F = \frac{GM x^3 m}{R^3 x^2} = \frac{GMm}{R^3}x\]
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