Patel, Kinal – Homework 19 – Due: Nov 13 2007, 7:00 pm – Inst: D Weathers
1
This
printout
should
have
11
questions.
Multiplechoice questions may continue on
the next column or page – find all choices
before answering.
The due time is Central
time.
001
(part 1 of 2) 10 points
A black hole is an object so heavy that neither
matter nor even light can escape the influence
of its gravitational field.
Since no light can
escape from it, it appears black.
Suppose a
mass approximately the size of the Earth’s
mass 6
.
32
×
10
24
kg is packed into a small
uniform sphere of radius
r
.
Use:
The
speed
of
light
c
=
2
.
99792
×
10
8
m
/
s
.
The universal gravitation
constant
G
= 6
.
67259
×
10

11
N m
2
/
kg
2
.
Hint:
The escape speed must be the speed
of light.
Based on Newtonian mechanics, determine
the limiting radius
r
0
when this mass (approx
imately the size of the Earth’s mass) becomes
a black hole.
Correct answer: 0
.
00938426 m.
Explanation:
Basic Concepts:
Gravitational energy
conservation
E
=

G m M
r
+
K .
At minimum escape velocity,
E
= 0 (the pro
jectile has just enough initial kinetic energy
to overcome the gravitational potential).
Solution:
Technically speaking, in a region
where gravity is extremely intense, Newton’s
mechanics cannot be used. Rather, one needs
to apply the “general theory of relativity”
developed by Albert Einstein. Knowing this
is the case, we still would like to see what
Newtonian mechanics tells us. Setting
v
esc
=
c
, the limiting radius is given by
1
2
m v
2
esc
=
1
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 Spring '07
 Weathers
 Mass, Work, General Relativity, Light, Correct Answer, patel

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