A force defined by acts on a particle at a given time . The factor which is dimensionless, if and are constants, is [2024]

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Solution

Q.
A force defined by $F = α t^{2} + β t$ acts on a particle at a given time $t$ . The factor which is dimensionless, if $α$ and $β$ are constants, is [2024]

1 $\frac{β t}{α}$

2 $\frac{α t}{β}$

3 $α β t$

4 $\frac{α β}{t}$

Ans.
(2)

$F = α t^{2} + β t$

Dimension of $α = \frac{[M L T^{- 2}]}{[T^{2}]} = [M L T^{- 4}]$

Dimension of $β = \frac{[M L T^{- 2}]}{[T]} = [M L T^{- 3}]$

Option a): $\frac{β t}{α} = \frac{[M L T^{- 3}] [T]}{[M L T^{- 4}]} = [M^{0} L^{0} T^{2}]$

Option b): $\frac{α t}{β} = \frac{[M L T^{- 4}] [T]}{[M L T^{- 3}]} = [M^{0} L^{0} T^{0}]$

Option c): $α β t = [M L T^{- 4}] [M L T^{- 3}] [T] = [M^{2} L^{2} T^{- 6}]$

Option d): $\frac{α β}{t} = \frac{[M L T^{- 4}] [M L T^{- 3}]}{[T]} = [M^{2} L^{2} T^{- 8}]$

Hence, option (b) is correct.

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