A gas bubble from an explosion under water, oscillates with a time period T proportional to p ^{a} d ^{b} E ^{c} , where p,d, and E are pressure, density and total energy respectively. Find the value of a, b and c.

given,

T ∝ P^{a }d^{b }E^{c}

^{ }= K P^{a }d^{b }E^{c }( K being some constant)

putting the dimensions of pressure,density,energy and time

[ T ] = K[M L^{-1} T^{-2}]^{a} [M L^{-3 }]^{b} [M L^{2 }T^{-2}]^{c}

= K [ M ]^{a+b+c }[ L ]^{-a-3b+2c}[ T ]^{-2a-2c}

compairing powers on LHS and RHS

a+b+c = 0 ...........................(1)

-a-3b+2c = 0 ............................(2)

-2a-2c = 1 ............................(3)

from (3)

a+c = -1/2

or, c = -1/2 - a

putting in (1) gives ** b= 1/2**

from (2)

-a -3(1/2) + 2(-1/2 - a) = 0

or, **a= -5/6**

from (3)

-2(-5/6) - 2c = 1

or, 5/3 -2c = 1

or, 2c = 5/3 - 1

or, **c = 1/3**

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