  Question

A solid sphere of uniform density and radius $$R$$ applies a gravitational force of attraction equal to $$F_1$$ on a particle placed at a distance 3R from the centre of the sphere. A spherical cavity of radius $$\dfrac{R}{2}$$ is now made in the sphere, as shown in the figure. The sphere with cavity now applies a gravitational force $$F_2$$ on the same particle. The ratio $$\dfrac{F_2}{F_1}$$ is A
950  B
4150  C
325  D
2225  Solution

The correct option is A $$\dfrac{41}{50}$$ From super position principle ,  $$F_{1}=F_{r}+F_{c}$$ Here $$F_{r}=$$ force due to remaining part $$=F_{2}$$ $$F_{c}=$$ force due to mass on the cavity $$F_{1}=\dfrac{GMm}{9R^{2}}$$ $$\displaystyle {F_{c}=\dfrac{G(\dfrac{M}{8})m}{(\dfrac{5}{2}R)^{2}}=\dfrac{GMm}{50R^{2}}}$$ $$=\displaystyle \succ F_{2}=F_{1}-F_{c}=\dfrac{GMm}{9R^{2}}-\dfrac{GMm}{50R^{2}}=\dfrac{41GMm}{450R^{2}}$$ $$\Rightarrow \displaystyle \dfrac{F_{2}}{F_{1}}=\dfrac{41}{50}$$ Physics

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