Gravitation with Extended Objects Thin Rod a point

Gravitation with Extended Objects

Thin Rod a point mass = m b mass of rod = M

Thin Rod As Point Mass b a point mass = m mass of rod = M Center of mass =

Comparison

Hollow Sphere Point mass P has mass m Sphere has mass M

The differential force of gravity on P from d. A is found by where M is the mass of the shell, d. M is the mass of d. A and r is the appropriate distance from P to the shell. It can be shown and therefore

Also it is known that where R is the radius of the shell and where r is the appropriate distance from P to the shell, so the differential force is found by Since the r 2 terms cancel out and all the other terms are exactly the same for d. A 1 and d. A 2 the net force on P is zero since the differential forces will lie on the axis in opposite directions. Logically this holds for every axis through the sphere containing P so the gravitational force on P is zero.

Solid Sphere For both a solid uniform sphere and a uniform shell it can be shown that for every differential mass d. M there is another mass d. M that is symmetric to the axis joining an external point P and the center of the sphere. This indicates that the gravitational force acts along the axis, and it can be further shown that the shell or sphere acts as a point mass at its center of mass.

Inside Solid Sphere If an object is located inside a solid uniform sphere of radius R at a distance r from the center of the sphere all of the mass except for that contained in the sphere defined by r can be considered to be a shell surrounding the object. This shell has a gravitational force of zero on the object, and only the sphere defined by r exerts a force on the object.

Graphs Graph of F vs. r for a hollow shell Graph of F vs. r for a solid sphere