Question

Two concentric spherical shells with uniformly distributed masses${\mathbf{M}}_{\mathbf{1}}$and${\mathbf{M}}_{\mathbf{2}}$are situated as shown inthe figure. Find the magnitude of the net gravitational force on a particleof massm, due to the shells, when theparticle is located at radial distance(a)a, (b)b, and (c)c.

Short answer

• The magnitude of the net gravitational force on a particle of mass m due to the shell, when the particle is located at radial distance is$G(M_1+M_2)ml{a}^2$
• The magnitude of the net gravitational force on a particle of mass m due to the shell, when the particle is located at radial distance b is$G{M}_{1}m/b^2$
• The magnitude of the net gravitational force on a particle of mass due to the shell, when the particle is located at radial distance c is 0

Step-by-step solution

The given data

Two concentric spherical shells with masses $M_1$and$M_2$

Understanding the concept of gravitational force

Using the formula for gravitational force, we can find the magnitude of the net gravitational force on a particle of mass due to the shell, when the particle is located at a radial distance.

Formula:

Gravitational force, $F=\frac{GMm}{r^2}$ (i)

a) Calculation of net force at radial distance

At point a,

$$\begin{gathered} r=a, \\ M=M_1+M_{2,} \\ m=m \end{gathered}$$

Substituting the values in equation (i), we get

$F=\frac{G(M_1+M_2)m}{a^2}$

Therefore, the magnitude of the net gravitational force on a particle of mass due to the shell, when the particle is located at radial distance a is$G\left(M_{1+}{M}_2\right)m/a^2$

b) Calculation of net force at radial distance b

At point a,

$$\begin{gathered} r=b, \\ M=M_1, \\ m=m \end{gathered}$$

Substituting the values in equation (i), we get

$F=\frac{G{M}_{1}m}{r^2}$

Therefore, the magnitude of the net gravitational force on a particle of mass due to the shell, when the particle is located at radial distance b is$G{M}_{1}m/b^2$

c) Calculation of net force at radial distance c

There is no other mass at point.

Therefore,

$F=0$

The magnitude of the net gravitational force on a particle of mass due to the shell, when the particle is located at radial distancec is 0