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(B) The gravitational force $F$ on a particle of mass $m$ at a distance $r$ from the center is given by $F = m \cdot E(r)$,where $E(r)$ is the gravita

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(B) The gravitational force $F$ on a particle of mass $m$ at a distance $r$ from the center is given by $F = m \cdot E(r)$,where $E(r)$ is the gravitational field intensity at that point.$1$. For $0 \le r \le R_1$: The particle is inside the cavity. By the shell theorem,the gravitational field inside a uniform spherical shell is zero. Thus,$F = 0$.$2$. For $R_1 \le r \le R_2$: The particle is within the material of the sphere. The effective mass $M_{eff}$ contributing to the force is the mass of the sphere of radius $r$ minus the mass of the cavity. $M_{eff} = \rho \cdot \frac{4}{3}\pi(r^3 - R_1^3)$,where $\rho = \frac{M}{\frac{4}{3}\pi(R_2^3 - R_1^3)}$. The force $F = \frac{G \cdot M_{eff} \cdot m}{r^2}$,which increases as $r$ increases.$3$. For $r > R_2$: The particle is outside the sphere. The sphere acts as a point mass $M$ at the center. Thus,$F = \frac{G M m}{r^2}$,which means $F \propto \frac{1}{r^2}$.Comparing these conditions,the graph that shows $F=0$ for $r  R_2$ is represented by option $(B)$.

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