Question

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Q64. A solid sphere of mass M and radius R has a spherical cavity of radius R/2 such that the centre of cavity is at a distance R/2 from the centre of the sphere. A point mass m is placed inside the cavity at a distance R/4 from the centre of sphere. The gravitational force on mass m is

$$\begin{gathered} \left(\mathrm{a}\right)\frac{11\mathrm{GMm}}{{\mathrm{R}}^2}\left(\mathrm{b}\right)\frac{14\mathrm{GMm}}{{\mathrm{R}}^2} \\ \left(\mathrm{c}\right)\frac{\mathrm{GMm}}{2{\mathrm{R}}^2}\left(\mathrm{d}\right)\frac{\mathrm{GMm}}{{\mathrm{R}}^2} \end{gathered}$$

Answer 1

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$$\begin{gathered} gravitationalfieldinthecavityis \\ E=\frac{4\pi G\rho }{3}(\left|{\overrightarrow{C_{1}C}}_2\right|)(C_{1}and{C}_{2}arethecentresofthesphereandthecavity) \\ \rho =\frac{M}{{\displaystyle \frac{4}{3}}\pi R^3}and{C}_1{C}_2=dis\tan cebetweenthecentres=\frac{R}{2} \\ E=\frac{4\pi G\times M}{3\times {\displaystyle \frac{4}{3}}\pi R^3}\left(\frac{R}{2}\right) \\ E=\frac{G\times M}{R^2}\left(\frac{1}{2}\right) \\ soforceactingonthechargeis \\ F=mE \\ F=\frac{G\times m\times M}{2R^2} \\ Regards \end{gathered}$$