The frequency (v) of an oscillating liquid dr | vedclass

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Question

(A) The dimensional formula for frequency $(v)$ is $[T^{-1}]$.The dimensional formula for radius $(r)$ is $[L]$.The dimensional formula for density $(

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$(-\frac{3}{2}, -\frac{1}{2}, \frac{1}{2})$

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(A) The dimensional formula for frequency $(v)$ is $[T^{-1}]$.The dimensional formula for radius $(r)$ is $[L]$.The dimensional formula for density $(\rho)$ is $[ML^{-3}]$.The dimensional formula for surface tension $(s)$ is $[MT^{-2}]$.Given the relation: $v = r^{a} \rho^{b} s^{c}$.Substituting the dimensions: $[M^{0}L^{0}T^{-1}] = [L]^{a} [ML^{-3}]^{b} [MT^{-2}]^{c}$.$[M^{0}L^{0}T^{-1}] = M^{b+c} L^{a-3b} T^{-2c}$.Comparing the powers of $M, L,$ and $T$ on both sides:For $T$: $-2c = -1 \Rightarrow c = \frac{1}{2}$.For $M$: $b + c = 0 \Rightarrow b = -c = -\frac{1}{2}$.For $L$: $a - 3b = 0 \Rightarrow a = 3b = 3(-\frac{1}{2}) = -\frac{3}{2}$.Thus,the values are $a = -\frac{3}{2}, b = -\frac{1}{2}, c = \frac{1}{2}$.

The frequency $(v)$ of an oscillating liquid drop may depend upon the radius $(r)$ of the drop,the density $(\rho)$ of the liquid,and the surface tension $(s)$ of the liquid as: $v = r^{a} \rho^{b} s^{c}$. The values of $a, b,$ and $c$ respectively are:

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