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A cavity of radius R/2 is made inside a solid sphere of radius R. The centre of the cavity is located at a distance R/2 from centre of the sphere.The gravitational force on a particle of mass 'm' at a distance R/2 from the centre of the sphere on the line joining both the centres of sphere and cavity is(opposite to centre of cavity.)?
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A cavity of radius R/2 is made inside a solid sphere of radius R. The ...
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A cavity of radius R/2 is made inside a solid sphere of radius R. The ...
Explanation:
To determine the gravitational force on a particle of mass 'm' at a distance R/2 from the center of the sphere on the line joining both the centers of the sphere and cavity, we need to consider the gravitational forces exerted by both the sphere and the cavity.
Gravitational Force due to the Solid Sphere:
The gravitational force exerted by a solid sphere on a particle outside the sphere is the same as if all its mass were concentrated at its center. Therefore, the gravitational force due to the solid sphere acting on the particle of mass 'm' at a distance R/2 from the center of the sphere is given by:
F_sphere = G * (m * M_sphere) / (R/2)^2
Where G is the universal gravitational constant and M_sphere is the mass of the solid sphere.
Gravitational Force due to the Cavity:
Inside the cavity, the gravitational force is zero since the mass inside the cavity cancels out due to the symmetry of the setup. However, outside the cavity, the gravitational force due to the cavity is non-zero and equal in magnitude but opposite in direction to the gravitational force due to the solid sphere.
The gravitational force due to the cavity acting on the particle of mass 'm' at a distance R/2 from the center of the sphere is given by:
F_cavity = -G * (m * M_cavity) / (R/2)^2
Where M_cavity is the mass of the cavity.
Net Gravitational Force:
Since the gravitational force due to the cavity is opposite in direction to the gravitational force due to the solid sphere, the net gravitational force on the particle is the difference between the two forces:
F_net = F_sphere + F_cavity
Substituting the expressions for F_sphere and F_cavity, we get:
F_net = G * (m * M_sphere) / (R/2)^2 - G * (m * M_cavity) / (R/2)^2
Simplifying the expression, we find:
F_net = G * (m * (M_sphere - M_cavity)) / (R/2)^2
Therefore, the gravitational force on the particle of mass 'm' at a distance R/2 from the center of the sphere on the line joining both the centers of the sphere and cavity (opposite to the center of the cavity) is given by the above equation.
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A cavity of radius R/2 is made inside a solid sphere of radius R. The centre of the cavity is located at a distance R/2 from centre of the sphere.The gravitational force on a particle of mass 'm' at a distance R/2 from the centre of the sphere on the line joining both the centres of sphere and cavity is(opposite to centre of cavity.)?
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