1
Unit 1
Kinematics
1.1
Dimensional Analysis
1.2
Vectors
1.3
Relative motion in one dimension
1.4
Relative motion in two dimensions
1.5
Motion with constant acceleration (1-D)
1.6
Projectile
1.1
Dimensional Analysis
The fundamental quantities used in physical descriptions are called dimensions. Length,
mass, and time are examples of dimensions. You could measure the distance between two
points and express it in units of meters, centimeters, or feet. In any case, the quantity would
have the dimension of length.
It is common to express dimensional quantities by bracketed symbols, such as [L], [M], and
[T] for length, mass, and time, respectively. Dimensional analysis is a procedure by which the
dimensional consistency of any equation may be checked. If I say,
x = at
, where
x
,
a
, and
t
represent the displacement, acceleration and time respectively.
Is it correct?
•
[
x
] = [L]
•
[
at
] = ( [L][T]
−
2
) ( [T] ) = [L][T]
−
1
So the dimension for the sides are not equal, i.e.
x = at
is invalid.
Now look at the equation
x = at
2
•
The L.H.S. is
x
, i.e. [L]
•
The R.H.S. is
at
2
, i.e. ( [L][T]
-2
) ( [T]
2
) = [L],
Hence the equation is dimensionally correct, but you should know that it is physically
incorrect. The correct equation is, as you should remember,
2
2
1
at
x
=
. The fraction
2
1
is a
constant and has no dimension, just like
π
.
Example
The period
P
of a simple pendulum is the time for one complete swing. How does
P
depend
on the mass
m
of the bob, the length
l
of the string, and the acceleration due to gravity
g
?