Gravitation Inside Earth — Question 1

Question 1

A uniform spherical shell has no preferred interior resting point

A thin uniform spherical shell has radius $R$ and mass $M$. Use the shell theorem: its gravitational field is zero everywhere strictly inside and equals a central point-mass field outside. A test particle is released from rest at $r=.7R$.

1. Describe its motion if the shell is fixed and no other forces act.

2. If it is instead given a small velocity, explain whether the interior equilibrium restores its original position.

Question 1 - Solution

A uniform spherical shell has no preferred interior resting point

(a) Derivation and calculation

The shell theorem gives $\vec g=\vec0$ everywhere strictly inside. Therefore $$m\ddot{\vec r}=m\vec g=\vec0,\qquad \vec r(t)=\vec r_0+\vec v_0t.$$ For release from rest, $$\boxed{\vec v(t)=\vec0,\qquad \vec r(t)=\vec r_0},\qquad |\vec r_0|=.7R.$$ The particle remains at its initial point.

(b) Result and interpretation

With a small imposed velocity, $$\boxed{\vec r(t)=\vec r_0+\vec v_0t},\qquad |\vec r(t)|<R.$$ A displacement alone leaves another equilibrium, since $$\delta\vec g=\vec0.$$ There is no restoring force; the particle drifts until it reaches the shell boundary.

Physical check

The interior motion is uniform only while it remains within the cavity and encounters no other force.