A gas bubble from an explosion under water oscillates with a time period T which depends upon static pressure p- density of water - and the total energyof explosion E- Find the expression for the time period T- where- k is a dimensionless constant A- T-k p-5 - 6-1 - 2 E1 - 3B- T-k p-4 - 7-1 - 2 E1 - 3C- T-k p-4 - 7-1 - 3 E1 - 2D- T-k p-5 - 6-1 - 2 E1 - 2

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A gas bubble ...

Question

A gas bubble from an explosion under water oscillates with a time period T which depends upon static pressure p, density of water $ρ$ and the total energy of explosion E. Find the expression for the time period T. (where, k is a dimensionless constant)

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Solution

The correct option is A
Time period
$T \propto p^{a} ρ^{b} E^{c}$
or, $T = k p^{a} ρ^{b} E^{c}$
k, is a dimensionless constant,
According to homogeneity of dimensions,
LHS =RHS
$∴ [T] = [M L^{- 1} T^{- 2}]^{a} [M L^{- 3}]^{b} [M L^{2} T^{- 2}]^{c}$
$[T] = [M^{a + b + c}] [L^{- a - 3 b + 2 c}] [T^{- 2 a - 2 c}]$
Comparing the powers, we obtain
$a + b + c = 0$
$- a - 3 b + 2 c = 0$
$- 2 a - 2 c = 1$
On solving, we get

$a = - \frac{5}{6}, b = \frac{1}{2}, c = \frac{1}{3} .$

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A gas bubble from an explosion under water oscillates with a time period T which depends upon static pressure p, density of water ρ and the total energy of explosion E. Find the expression for the time period T. (where, k is a dimensionless constant)

A gas bubble from an explosion under water oscillates with a time period T which depends upon static pressure p, density of water $ρ$ and the total energy of explosion E. Find the expression for the time period T. (where, k is a dimensionless constant)