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

Correct answer: A. (-(3)/(2),-(1)/(2) ; (1)/(2)) aligned & v= ^ a ^ b ^ c [v]=[r]^ a [ ]^ b [s]^ c & [T^ -1 ]=[L^1]^ a [M^1 L^ -3 ]^ b [ MLT L ]^ -2 & T^ -1 =M^ b+c · L^ a-3 b · T^ -2 c & b+c=0, a-3 b=0,-2 c=-1 & c=(1)/(2), b=-(1)/(2), a-3 b=0 ; & a+(3)/(2)=0 a=-(3)/(2) aligned

BITSAT 2023Paper 2

Exam: BITSAT
Year: 2023
Paper: Paper 2
Subject: Physics
Chapter: Units and Dimensions

BITSAT 2023 · Paper 2 · Physics · Units and Dimensions

The frequency (v) of an oscillating liquid drop may depend upon...

The frequency

(v)

of an oscillating liquid drop may depend upon radius

(r)

of the drop, density

(\rho)

of 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

Options

A $\left(-\frac{3}{2},-\frac{1}{2} ; \frac{1}{2}\right)$
B $\left(\frac{3}{2},-\frac{1}{2}, \frac{1}{2}\right)$
C $\left(\frac{3}{2}, \frac{1}{2} ;-\frac{1}{2}\right)$
D $\left(-\frac{3}{2}, \frac{1}{2}, \frac{1}{2}\right)$

Correct answer

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

Step-by-step solution

\begin{aligned} & v=\mathrmr^{\mathrm{a}} \rho^{\mathrm{b}} \mathrms^{\mathrm{c}} \Rightarrow[v]=[\mathrm{r}]^{\mathrm{a}}[\rho]^{\mathrm{b}}[\mathrm{s}]^{\mathrm{c}} \\ & \Rightarrow \quad\left[\mathrm{T}^{-1}\right]=\left[\mathrm{L}^1\right]^{\mathrm{a}}\left[\mathrm{M}^1 \mathrm{ L}^{-3}\right]^{\mathrm{b}}\left[\frac{\mathrm{MLT}}{\mathrm{L}}\right]^{-2} \\ & \Rightarrow \mathrm{T}^{-1}=\mathrm{M}^{\mathrm{b}+\mathrm{c}} \cdot \mathrm{L}^{\mathrm{a}-3 \mathrm{ b}} \cdot \mathrm{T}^{-2 c} \\ & \Rightarrow \mathrm{b}+\mathrm{c}=0, \mathrm{a}-3 \mathrm{ b}=0,-2 \mathrm{c}=-1 \\ & \Rightarrow \quad \mathrm{c}=\frac{1}{2}, \mathrm{ b}=-\frac{1}{2}, \mathrm{a}-3 \mathrm{ b}=0 ; \\ & \mathrm{a}+\frac{3}{2}=0 \Rightarrow \mathrm{a}=-\frac{3}{2}\end{aligned}

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