Simple Nature - Light and Matter

k/The gravitational energy
of a massmat a distancesfrom
the center of a hollow spherical
shell of mass.

and since sinθdθoccurs in the integral, the easiest path is to sub-

stitute for it, and get everything in terms ofrand dr:

U=−

2 πGρbhm

s

∫s+b

s−b

dr

=−

4 πGρb^2 hm

s

=−

GMm

s

This was all under the assumption that massmwas on the outside

of the shell. To complete the proof, we consider the case where it’s

inside. In this case, the only change is that the limits of integration

are different:

U=−

2 πGρbhm

s

∫b+s

b−s

dr

=− 4 πGρbhm

=−

GMm

b

The two results are equal at the surface of the sphere,s=b,

so the constant-energy part joins continuously onto the 1/spart,

and the effect is to chop off the steepest part of the graph that we

would have had if the whole massMhad been concentrated at its

center. Dropping a massmfrom A to B in figure k releases the

same amount of energy as if massMhad been concentrated at its

center, but there is no release of gravitational energy at all when

moving between two interior points like C and D. In other words,

the internal gravitational field is zero. Moving from C to D brings

massmfarther away from the nearby side of the shell, but closer

to the far side, and the cancellation between these two effects turns

out to be perfect. Although the gravitational field has to be zero

at the center due to symmetry, it’s much more surprising that it

cancels out perfectly in the whole interior region; this is a special

mathematical characteristic of a 1/rinteraction like gravity.

Newton’s apple example 17

Over a period of 27.3 days, the moon travels the circumference

of its orbit, so using data from Appendix 4, we can calculate its

speed, and solve the circular orbit condition to determine the

strength of the earth’s gravitational field at the moon’s distance

from the earth,g = v^2 /r = 2.72× 10 −^3 m/s^2 , which is 3600

times smaller than the gravitational field at the earth’s surface.

The center-to-center distance from the moon to the earth is 60

times greater than the radius of the earth. The earth is, to a very

good approximation, a sphere made up of concentric shells, each

with uniform density, so the shell theorem tells us that its external

gravitational field is the same as if all its mass was concentrated

104 Chapter 2 Conservation of Energy