If force (F), velocity (V) and time (T) are taken as fundamental units, then the dimensions of mass are [2014]

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Solution

Q.
If force (F), velocity (V) and time (T) are taken as fundamental units, then the dimensions of mass are [2014]

1 $[F V T^{- 1}]$

2 $[F V T^{- 2}]$

3 $[F V^{- 1} T^{- 1}]$

4 $[F V^{- 1} T]$

Ans.
(4)

Let mass $m \propto F^{a} V^{b} T^{c}$ or $m = k F^{a} V^{b} T^{c}$ ...(i)

where $k$ is a dimensionless constant and $a, b$ and $c$ are the exponents.

Writing dimensions on both sides, we get

$[M L^{0} T^{0}] =$ ${[M L T^{- 2}]}^{a}$ ${[L T^{- 1}]}^{b}$ ${[T]}^{c}$

$[M L^{0} T^{0}] = [M^{a} L^{a + b} T^{- 2 a - b + c}]$

Applying the principle of homogeneity of dimensions, we get

$a = 1$ ...(ii), $a + b = 0$ ...(iii), $- 2 a - b + c = 0$ ...(iv)

Solving eqns. (ii), (iii) and (iv), we get $a = 1, b = - 1, c = 1$

From eqn. (i), $[m] = [F V^{- 1} T]$

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