First slide

Dimensions of physical quantities

Question

The SI unit of energy is $J = k g m^{2} s^{- 2},$ that of speed v is $m s^{- 1}$ and of acceleration a is $m s^{- 2}$ which one of the formula for kinetic energy given below is correct on the basis of dimensional arguments.
[Given m stands for the mass of body]

Easy

A

$K = m^{2} v^{2}$
B

$K = m a$
C

$K = \frac{1}{2} m v^{2} + m a$
D

$K = \frac{1}{2} m v^{2}$

Solution

Every correct formula or equation must have the same dimensions on both sides of the equation. Also, only quantities with the same physical dimensions can be added or
subtracted. The dimensions of the quantity on the right side for (a) is
$K = m^{2} v^{2}$
Putting, $m = [M], v = [L T^{- 1}]$
$∴ \dim e n s i o n s a r e [M^{2} L^{2} T^{- 2}]$
for (b), K =ma
putting $m = [M], a = [L T^{- 2}]$
$∴ \dim e n s i o n s a r e [M L T^{- 2}]$
option (c) has no proper dimensions,
option (d), $K = \frac{1}{2} m v^{2},$ putting m = [M]
and $V = [L T^{- 1}],$ we have
$K = [M] {[L T^{- 1}]}^{2} = [M L^{2} T^{- 2}]$
in units it is written as $k g m^{2} s^{- 2}$

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