Physics 341: Problem Set #1 Solutions
1.
(a)
Use dimensional analysis to derive a relationship between the total mass of a
gravitationally bound system
M
, its typical size
R
, and the typical speed
v
of its
components.
We can start by writing the dimensions for the quantities involved, as well as
Newton’s gravitational constant (because gravity is holding the system together).
gravity
G
[
M

1
L
3
T

2
]
mass
M
[
M
]
size
R
[
L
]
speed
v
[
L T

1
]
If we solve for the speed, some trial and error quickly yields
v
∼
GM
R
1
/
2
Or similarly, we could have solved for the mass or the size
M
∼
v
2
R
G
R
∼
GM
v
2
(b)
A globular cluster has about
10
6
stars in a sphere of radius 10 pc.
Using your
analysis above, estimate the speed with which the stars move inside the cluster, in
units of
km s

1
. You will need to make an assumption about the average star; be
sure to state any assumptions explicitly.
We need to know the mass of the cluster, so we’ll have to make an assumption.
If we assume the stars are similar to the Sun on average then for the total mass
of the cluster we have
M
= 10
6
M
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 Fall '08
 Keeton
 Mass, Supernova, White dwarf, km s−1

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