Question

A thin spherical shell has radius 4.00 m (A) and mass 20.0 kg (A). It is concentric with a second thin spherical shell that has radius 6.00 m (B) and mass 40.0 kg (B). What is the net gravitational force that the two shells exert on a point mass of 0.0200 kg that is a distance r from the common center of the two shells, for: a) r = 2.00 m (inside both shells) b) r = 5.00 m (in the space between the two shells) c) r = 8.00 m (outside both shells)

Answer 1

a) Inside both shells r = 2.00 m, the gravitational force F_a on the point mass of 0.0200 kg is determined by the mass of the inner shell (A) and the formula F_a = G . (0.0200 kg) . (20.0 kg)} / {(2.00 m)^2}.

b) In the space between the two shells (r = 5.00 m), the net gravitational force (F_between) on the point mass is the sum of the forces from both shells, given by F_between = {G . (0.0200 kg) . (20.0 kg)} / {(5.00 m)^2} + {G . (0.0200 kg) . (40.0 kg) / {(5.00 m)^2).

c) Outside both shells (r = 8.00 m), the gravitational force (F_b) is determined by the mass of the outer shell (B) and follows the formula \(F_b = {G . (0.0200 kg) . (40.0 kg)} / {(8.00 m)^2}\).

Step-by-step explanation:

Net Gravitational Force Between Spherical Shells

To calculate the net gravitational force that two spherical shells exert on a point mass, we will use Newton's law of universal gravitation. According to this law, the gravitational force (F) between two masses is given by the equation F = G * (m1 * m2) / r^2, where G is the gravitational constant, m1 and m2 are the masses, and r is the distance between their centers.

However, due to the shell theorem, the gravitational force inside a uniform spherical shell of mass is zero. This theorem will significantly impact the way we approach each part of the question.

a) For r = 2.00 m, inside both shells, the gravitational force on the point mass is zero due to the shell theorem, since all the mass of both shells can be considered to be concentrated at points outside the radius where the point mass is located.

b) For r = 5.00 m, the point mass is only influenced by shell A (inner shell), as shell B is still outside its radius. The net gravitational force is hence F = G * (mass of shell A * mass of the point) / r^2. Here, we ignore shell B according to the shell theorem.

c) For r = 8.00 m, the point mass is outside both shells, so both contribute to the gravitational force. The net force is the sum of the forces due to both shells, each calculated by the equation F = G * (mass of shell * mass of the point) / r^2.