First slide

Dimensions of physical quantities

Question

If an unknown quantity, $ϕ = \frac{m a}{α} \log (1 + \frac{α l}{m a}) .$ where m = mass, a = acceleration and l = length.

Moderate

A

$[M L T^{- 2}]$
B

$[M T^{- 2}]$
C

$[M^{0} L T^{0}]$
D

$[M L^{- 3}]$

Solution

Logarithm has no dimensions.
$\begin{array}{l} ∴ 1 + \frac{α l}{m a} = dimensionless \\ ∴ \frac{α l}{m a} = [M^{0} L^{0} T^{0}] \\ \Rightarrow α = \frac{m a}{l} = \frac{[M L T^{- 2}]}{[L]} = [M T^{- 2}] \end{array}$
$∵ Dimensions of ϕ = Dimensions of \frac{m a}{α}$
$= \frac{[M L T^{- 2}]}{[M T^{- 2}]} = [L] = [M^{0} L T^{0}]$

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