he minimum work done by an external agent in shifting a particle of mass m inside a thin spherical shell of mass M and radius R is:
Question
he minimum work done by an external agent in shifting a particle of mass m inside a thin spherical shell of mass M and radius R is:
Solution
The work done by an external agent in shifting a particle of mass m inside a thin spherical shell of mass M and radius R is given by the formula:
W = ∫ F dx
where F is the force exerted by the external agent and dx is the displacement of the particle.
The force exerted by the external agent must be equal and opposite to the gravitational force exerted by the shell on the particle. The gravitational force is given by:
F = GmM/r^2
where G is the gravitational constant and r is the distance from the center of the shell to the particle.
The displacement dx is equal to the change in the particle's position, which is equal to the radius R of the shell.
Substituting these values into the formula for work gives:
W = ∫ GmM/r^2 dx
This integral is evaluated from r = R (the initial position of the particle) to r = 0 (the final position of the particle). The result is:
W = GmM/R
This is the minimum work done by an external agent in shifting a particle of mass m inside a thin spherical shell of mass M and radius R.
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