A very small round ball is located near a large solid sphere of uniform density. The force that the large sphere exerts on the ballA) is exactly the same as it would be if all the mass of the sphere were concentrated at the center of the sphere. B) is approximately the same as it would be if all the mass of the sphere were concentrated at the center of the sphere. C) is independent of the mass of the ball. D) is independent of the mass of the sphere. E) can only be calculated using calculus.

B) is approximately the same as it would be if all the mass of the sphere were concentrated at the center of the sphere.

Explanation:

Newton’s Law of Universal Gravitation says that if we have two point masses

m and M separated by a distance r, then the mutual force exerted on each is

given by

F = G

mM

r

2

,

where the universal constant is G has approximate value1

G ≈ 6.67 × 10−11N · m2/kg2.

Sometimes, it’s more convenient to measure instead the gravitational field

E resulting from a point with mass M; measured in units of Newtons per kilogram

it measures the force on a point mass (of 1 kg) placed in this field. Therefore,

E will be directed radially inward toward the initial point mass and have a field

strength

E = ||E|| =GMr2,

at a distance r (meters) away from the point with mass M.

For an extended massive object with mass M, not concentrated at a point, the determination of the resulting gravitational field at a given point requires that the contributions of each component particle of mass dM are integrated into a final answer. Newton’s Shell Theorem states essentially two things and has a very important consequence. First of all, it says that the gravitational field outside a spherical shell having total mass M is the same as if the entire mass M is concentrated at its center (center of mass). Secondly, it says that for the same sphere the gravitational field inside the spherical shell is identically 0.

As a consequence of Newton’s shell method, one can conclude immediately that for a spherical homogeneous solid having mass M, the resulting gravitational field is again the same as if the entire mass were concentrated at a point. A somewhat more esoteric consequence is that if the spherical homogeneous object has radius R, then the gravitational field inside the object as a distance r < R from the center is the same as if total mass within a distance r from the center were concentrated at the object’s center.

b. a lightning bug, plug of grass, and a worm