First slide

Dimensions of physical quantities

Question

Force F is given in terms of time t and distance x by $F = A \sin (C t) + B \cos (D x)$ . Then the dimensions of A/B and C/D are

Moderate

A

$[M^{0} L^{0} T^{0}], [M^{0} L^{0} T^{- 1}]$
B

$[M L T^{- 2}], [M^{0} L^{- 1} T^{0}]$
C

$[M^{0} L^{0} T^{0}], [M^{0} L T^{- 1}]$
D

$[M^{0} L^{1} T^{- 1}], [M^{0} L^{0} T^{0}]$

Solution

$\begin{array}{l} \frac{A}{B} = \frac{Force}{Force} = [M^{0} L^{0} T^{0}] \\ C t = angle \Rightarrow C = \frac{Angle}{Time} = \frac{1}{T} = [T^{- 1}] \\ D x = angle \Rightarrow D = \frac{Angle}{Distance} = \frac{1}{L} = [L^{- 1}] \\ ∴ \frac{C}{D} = \frac{T^{- 1}}{L^{- 1}} = [M^{0} L T^{- 1}] \end{array}$

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