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Definition: The Dimension is the qualitative nature of a physical quantity (length, mass, time).
Square brackets denote the dimension or units of a physical quantity:
quantity | dimension | SI units |
area | [A] = L ^{2} | m ^{2} |
volume | [V]=L ^{3} | m ^{3} |
velocity | [v] = L/t | m/s |
acceleration | [a] = L/t ^{2} | m/s ^{2} |
mass | [m] = M | kg |
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.
Note:
Dimensional analysis can't give numerical factors. For Example:
The distance (x) travelled by a car in a given time (t) , starting
from rest and moving with constant acceleration (a) is given by,
x = at^{ 2}. We can check this equation with dimensional analysis:
l.h.s. [x] | = | L | |
r.h.s. | = | [a][t^{ 2}] = t ^{2} = L. |
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