Assume the given relation

$u=r^a\rho^b s^c$

where:

• $u$ = frequency
• $r$ = radius
• $\rho$ = density
• $s$ = surface tension

Write dimensions of each quantity

• Frequency: $[u]=T^{-1}$
• Radius: $[r]=L$
• Density: $[\rho]=ML^{-3}$
• Surface tension: $[s]=\frac{\text{force}}{\text{length}}=\frac{MLT^{-2}}{L}=MT^{-2}$

Substitute dimensions into the relation

$[u]=[r]^a[\rho]^b[s]^c$

$T^{-1}=L^a(ML^{-3})^b(MT^{-2})^c$

Expanding:

$T^{-1}=L^a M^b L^{-3b} M^c T^{-2c}$

$T^{-1}=M^{b+c}L^{a-3b}T^{-2c}$

Compare powers of $M$, $L$, and $T$

For mass $M$: $b+c=0$

For length $L$: $a-3b=0$

For time $T$: $-2c=-1$ $c=\frac{1}{2}$

Find $b$ and $a$

From $b+c=0$, $b=-\frac{1}{2}$

From $a-3b=0$, $a=3b=3\left(-\frac{1}{2}\right)=-\frac{3}{2}$

Final values

$a=-\frac{3}{2},\quad b=-\frac{1}{2},\quad c=\frac{1}{2}$

Match with options

This corresponds to:

Option B

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