First slide

Dimensions of physical quantities

Question

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

Moderate

A

$T = k p^{- 5 / 6} ρ^{1 / 2} E^{1 / 3}$
B

$T = k p^{- 4 / 7} ρ^{1 / 2} E^{1 / 3}$
C

$T = k p^{- 5 / 6} ρ^{1 / 2} E^{1 / 2}$
D

$T = k p^{- 4 / 7} ρ^{1 / 3} E^{1 / 2}$

Solution

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 -3b + 2c = 0
-2a -2c =1
On solving, we get
$a = \frac{5}{6}, b = \frac{1}{2}, c = \frac{1}{3}$

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