Chapter A  Problems
Blinn College  Physics 2425  Terry Honan
Problem A.1
Suppose that
x
is a length,
t
is a time and
a
is an acceleration. To get a general expression for
x
as a function of both
t
and
a,
we
choose the general form:
x
= k
a
m
t
n
where
k
is a dimensionless constant. For this expression to be dimensionally correct what must
m
and
n
be?
Solution to A.1
Since
x
is a length,
t
is a time and
a
is an acceleration we get:
@
x
D
=
L ,
@
t
D
=
T and
@
a
D
=
L
T
2
. A dimensionless constant has dimen
sion 1, so
@
k
D
=
1. Equating the dimensions of both sides of our general expression gives:
@
x
D
=
@
k
D @
a
D
m
@
t
D
n
ï
L
1
T
0
=
1
ÿ
K
L
T
2
O
m
μ
T
n
=
L
m
μ
T
n

2
m
When we equate the powers of L and T we get two equations:
1
=
m
and 0
=
n

2
m
which has the unique solution m = 1 and n = 2 .
Problem A.2
Three fundamental constants G, c and
—
have dimensions:
[
G
] =
L
3
M
ÿ
T
2
, [
c
] =
L
T
and [
—
] =
M
L
2
T
.
(a) What must
m
,
n
and
p
be to make
L
0
a length, when
L
0
=
G
m
ÿ
c
n
ÿ
—
p
.
(b)
—
, called "h bar", is a rescaled version of Plank's constant
h
; it is usually also called Plank's constant.
—
=
h
2
p
=
1.054
μ
10

34
J
ÿ
s
Using this and the values of
G
and
c
given in the book evaluate
L
0
.
Comment:
G
is
Newton's Gravitational constant
,
c
is the
speed of light
which plays a central role in Relativity and
—
is called
Plank's
constant
, which appears in Quantum theory.
L
0
is known as the
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 Spring '09
 Linear Equations, Acceleration, General Relativity, Acre, Planck units

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