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Dimensional Analysis
Definition:
The Dimension is the qualitative nature of
a physical quantity (length, mass, time).
brackets [ ] denote the dimension or units of a physical quantity:
Quantity
Length
Mass
Time
Dimension
[L]=L
[M]=M
[T]=T
Area
[A] = L2
Volume
[V]=L3
velocity
[v]= L/T
Acceleration
force
[a] = L/T2
[f]=M L/T2
[Q]
- If an equation is dimensionally correct, does this
mean that the equation must be true?
- If an equation is not dimensionally correct, does
this mean that the equation cannot be true?
Idea:
Dimensional analysis can be used to derive or check formulas by treating dimensions
as algebraic quantities.
Quantities can be added or subtracted only if they have the same dimensions, and
quantities on two sides of an equation must have the same dimensions.
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Example :
Using the dimensional analysis check this equation is correct:
x = ½ at 2
where:
x: is the distance,
a: is the acceleration and
Solution left side
righ side
t: is the time.
[x]  L
1
L
[ at 2 ]  2  T 2  L
2
T
This equation is correct because the dimension of the left and right side of the equation have
the same dimensions.
Dr. Loai Afana
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Example :
Show that the expression
x = vt +1/2 at2
is dimensionally correct ,
Where:
x is coordinate and has unit of length,
v is velocity,
a is acceleration, and
t
is the time.
The left side
The right side
[ x]  L
L
L 2
[vt  1 2 at ]  T  1 / 2 2 T
T
T
[ L]  [ L]
2
Dr. Loai Afana
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Example:
Which of the following equations are dimensionally correct?
Where: (x, y, m) is unit of length, (v) is velocity, ( a) is acceleration
(a ) v f  v i  ax
(b ) y  (2m ) cos(kx ), where k  2m 1
Solution :
(a) This is incorrect:
L L L
L L2
  .L   2
T T T2
T T
Dr. Loai Afana
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Example:
The position of a particle moving under uniform acceleration is some function of time
and the acceleration. Suppose we write this position:
x = kam tn
where , k is a dimensionless constant.
Use the dimensional analysis to determine the power n and m.
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Example:
A particle moving in circle of radius r with uniform velocity v,
Suppose that the acceleration (a) of the particle is proportional to the rn and vm.
Use the dimensional analysis to determine the power n and m.
Dr. Loai Afana
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