Assume that a tunnel is dug through the Earth | vedclass

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(B) When a particle of mass $m$ is at a distance $r$ from the center of the Earth,the gravitational force acting on it is due only to the mass of the

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$\left( \frac{4\pi}{3} mG\rho \right) r$

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(B) When a particle of mass $m$ is at a distance $r$ from the center of the Earth,the gravitational force acting on it is due only to the mass of the Earth contained within a sphere of radius $r$. Let this mass be $M'$.The volume of this inner sphere is $V' = \frac{4}{3} \pi r^3$.Since the density $\rho$ is uniform,the mass $M'$ is given by $M' = \rho V' = \rho \left( \frac{4}{3} \pi r^3 \right)$.The gravitational force $F$ on the particle of mass $m$ at distance $r$ is given by Newton's law of gravitation:$F = \frac{G M' m}{r^2}$Substituting the value of $M'$:$F = \frac{G m}{r^2} \left( \rho \cdot \frac{4}{3} \pi r^3 \right)$Simplifying the expression:$F = \left( \frac{4}{3} \pi G \rho \right) m r$

Assume that a tunnel is dug through the Earth from the North Pole to the South Pole and that the Earth is a non-rotating,uniform sphere of density $\rho$. The gravitational force on a particle of mass $m$ dropped into the tunnel when it reaches a distance $r$ from the center of the Earth is:

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