Question

The air bubble formed by explosion inside water performed oscillation with time period $T$ that is directly proportional to $p^a{d}^b{E}^c$ , where $p$ is the pressure, $d$ is the density and $E$ is the energy due to explosion. The values of $a,b$ and $c$ will be

• A. $-5/6,1/2,1/3$
• B. $5/6,1/3,1/2$
• C. $5/6,1/2,1/3$
• D. None of the above

Answer 1

Answer: (A)

Given that,
time period, $T\propto P^a{d}^b{E}^c\dots \left(i\right)$
The dimensions of these quantities are given as
$p=\left[M{L}^{-1}{T}^{-2}\right]$
$d=\left[M{L}^{-3}\right]$
$E=\left[M{L}^2{T}^{-2}\right]$
In Eq. $\left(i\right)$, on writting the dimensions on both sides
$\left[M^0{L}^{0}T\right]\propto {\left[M{L}^{-1}{T}^{-2}\right]}^a{\left[M{L}^2{T}^{-2}\right]}^c$
$\Rightarrow \left[M^0{L}^{0}T\right]\propto \left[M^{a+b+c}{L}^{-a-3b+2c}{T}^{-2a-2c}\right]$
On comparison the powers of $M,L,T$ on both sides.
$\Rightarrow a+b+c=0\dots \left(ii\right)$
$-a-3b+2c=0\dots \left(iii\right)$
$-2a-2c=1\dots \left(iv\right)$
Now from Eq. $\left(ii\right)$,
$b=-\left(a+c\right)$
On putting the value of $b$ in Eq.$\left(iii\right)$,
$-a+3\left(a+c\right)+2c=0$
or $-a+3a+3c+2c=0$
or $2a+5c=0\dots \left(v\right)$
On adding Eqs.$\left(iv\right)$ and $\left(v\right)$.
$3c=1$
$\Rightarrow c=\frac{1}{3}$
On putting the value of c in Eq. $\left(v\right)$
$2a+\frac{5}{3}=0$
or $2a=-\frac{5}{3}$
or $a=-\frac{5}{6}$
On putting the values of a and c in Eq. $\left(ii\right)$, we have
$-\frac{5}{6}+b+\frac{1}{3}=0$
$\Rightarrow b=\frac{5}{6}-\frac{1}{3}$
$\Rightarrow b=\frac{5-2}{6}=\frac{3}{6}$
$\Rightarrow b=\frac{1}{2}$
Hence value of a, b and c are $-\frac{5}{6},\frac{1}{2}$ and $\frac{1}{3}$