MATH 164, Spring 2001 Due Date: Name(s):
Honors Project 3: Improper Integrals and Black Holes
An integral of the form
rE
0
1
r2
dr
is an improper integral. A black hole is an object associated with a star which has collapsed on itself. In
this project we investigate the connection between improper integrals and black holes.
The weight w of an object near the surface of the earth is often taken, by definition, to be w = mg where
m is the mass of the object and g is a constant. Weight is a result of the force of gravity. (In English units,
w is measured in pounds, m in slugs, and g = 32 ft/sec2
.) The work W done in lifting an object of constant
weight (or mass) a distance h above the surface of the earth is thus approximately
W = weight  distance = wh = mgh. (1)
(If the weight (or mass) of the object varied with distance h above the surface of the earth1
w = w(h) = m(h)gh
then
W =
h
0
w dh =
h
0
m(h)g dh = g
h
0
m(h) dh.
In the special case that m does not depend on h,
W = g
h
0
m dh = mgh,
in agreement with (1).)
The above analysis assumes the object is near the surface of the earth. It is based on an approximation
to the Universal Law of Gravitation, which states that the attraction between two objects of masses m1 and
m2 which are r units apart is
F =
Gm1m2
r2
(2)
where G is a universal gravitational constant. For an object of constant mass m which is a distance h above
the surface of the earth,
F =
GmEm
(rE + h)2

GmEm
r2
E
=
GmE
r2
E
m = mg
where g = GmE/r2
E, mE is the mass of the earth, and rE is the radius of the earth. (This approximation is
justified by the fact that h is generally much smaller than rE  3963 mi, so rE + h  rE.)
Exercise
1. Avoiding approximations, find the work W done in lifting an object of constant weight (or mass) a distance
h away from the surface of the earth.
1This would happen, for example, if the object being lifted was a bucket full of water with a hole through which water
leaked.
The assumption you made in doing Exercise 1 was that the radius rE of the earth is fixed, and the object
is lifted a distance h away from the surface of the earth. Let us now change our point of view: instead of
lifting the object, consider what would happen if we shrank the earth while maintaining its mass. Let us
assume, for example, that we level all mountains and valleys on the earth, and eliminate all hollow spaces
(such as caves) within the earth.
Exercise
2. If the radius of the shrunken earth is rS, find the work done in lifting an object of constant weight (or
mass) m from the surface of the shrunken earth to the radius rE of the "old" earth.
Let us now assume that we shrink the earth further and further, all the while maintaining its mass. (Such
a process would be possible if, for example, we were to alter the molecular structure of matter, making
component materials denser and denser.)
Exercise
3. What is the limit of the work done in lifting an object of constant weight (or mass) m from the surface
of a shrunken earth to the radius rE of the "old" earth as the radius rS of the shrunken earth approaches 0.
4. Interpret your answer to Exercise 3 in terms of physics.
A black hole is an object associated with a star which has (for whatever reason) collapsed on itself. If, in
the process of collapsing, the star maintains its mass then the resulting black hole exerts tremendous forces
on objects close to it: forces so large that not even light can escape its gravitational pull. From the point of
view of mathematics, this is because an associated improper integral
rE
0
1
r2
dr = lim
rS 0
rE
rS
1
r2
dr
is infinite.