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, $T \propto P^{a}d^{b}E^{c} \ldots\left(i\right)$
The dimensions of these quantities are given as
$p=\left[ML^{-1}T^{-2}\right]$
$d=\left[ML^{-3}\right]$
$E=\left[ML^{2}T^{-2}\right]$
In Eq. $\left(i\right)$, on writting the dimensions on both sides
$\left[M^{0}L^{0}T\right]\propto\left[ML^{-1}T^{-2}\right]^{a}\left[ML^{2}T^{-2}\right]^{c}$
$\Rightarrow \left[M^{0}L^{0}T\right]\propto\left[M^{a+b+c}L^{-a-3b+2c} T^{-2a-2c}\right]$
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 $\frac{1}{3}$