2.6.E—Describe the gravitational force exerted on an object by a uniform spherical distribution of mass

Describe the gravitational force exerted on an object by a uniform spherical distribution of mass.

• The net gravitational force exerted on an object by a uniform spherical distribution of mass is the sum of the individual forces from small differential masses that comprise the distribution.
• Newton’s shell theorem describes the net gravitational force exerted on an object by a uniform spherical shell of mass.
  • i. The net gravitational force exerted on an object inside a thin spherical shell is zero.
  • ii. The net gravitational force exerted on an object outside a thin spherical shell can be determined by treating the shell as a single massive object located at the center of the shell.
  • iii. An object inside a sphere of uniform density experiences a net gravitational force from only a partial mass of the sphere.
  • iv. The partial mass of a sphere that contributes to the net gravitational force exerted on an object within that sphere is the portion of the sphere’s mass located a distance less than or equal to the object’s distance from the center of the sphere and can be calculated using the density of the sphere. Derived equation:
• The gravitational force exerted on an object within a uniform sphere can be shown to be proportional to the object’s distance from the sphere’s center. Derived equation: Fk rg,p artial pa rtial=− BOUNDARY STATEMENT AP Physics C: Mechanics does not expect students to mathematically prove or derive Newton’s shell theorem.