Problem .5: A gas bubble from an explosion under water, oscillates with a period proportional to \(
\text{P}^a \rho ^\text{b} \text{E}^{\text{c }}
\) where P is the static pressure, \(\rho\) is the density of water and E is the total energy of the explosion. Find the values of a, b, and c.
Sol. Let, \(
\text{T}\alpha \text{P}^a \rho ^\text{b} \text{E}^{\text{c }}
\)
Dimensional formula of
\(
\text{p} \to \text{ML}^{ - 1} \text{T}^{ - 2} ;\rho \to \text{ML}^{ - 3}
\) ; \(
\text{E} \to \text{[M L}^2 \text{T}^{ - 2} \text{]}
\)
\(
\text{T}\alpha \left( {\text{M L}^{ - 1} \text{T}^{ - 2} } \right)^a \left( {\text{ML}^{ - 3} } \right)^\text{b} \left( {\text{M L}^2 \text{T}^{ - 2} } \right)^{\text{c }}
\)
Comparing powers of M, L and T on both sides
a+b+c=0 ------(1)
-a-3b+2c=0 ------(2)
-2a-2c = 1 -----(3)
From (3) and (1):
\(
\text{a + c = - }\frac{1}
{2}
\) and b=\(\frac {1}
{2}\) = \(
\text{ - }\frac{1}
{2}
\); \(
\frac{{ - \text{a + 2c = 3} \times \frac{1}
{2}}}
{{\text{3c = 1}}}
\)
\(
\Rightarrow \text{c = }\frac{1}
{3}
\) and a=\(\text{ - }\frac{1}
{3}\text{ - }\frac{1}
{2} = \frac{{ - 5}}
{6}
\)
:. The values of a, b, c are respectively, \(
\frac{{ - 5}}
{6},\frac{1}
{2},\frac{1}
{3}
\)