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# A gas bubble from an explosion under water oscillates with a period proportional to p to the power a d to the b and E to the pow

A gas bubble from an explosion under water oscillates with a period proportional to p to the power a d to the b and E to the power c  where p is

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Given time period T

$T\alpha p^ad^bE^c$
$T=kp^ad^bE^c......(1)$
where K is a constant of proportionality and dimensionless quantity.

Inserting the dimensions of Time, pressure, density and Energy in equation (1) we get

$[T]=[ML^{-1}T^{-2}]^{a}[ML^{-3}]^b[ML^2T^{-2}]^c$Equating powers of M,L,andT on both sides we get

0=a+b+c .....(2)
0=–a–3b+2c .....(3)
1=–2a–2c ......(4)

Solving these equations

From (4)

a+c=−1/2 .....(5)

Inserting this value in (2)

0=b−(1/2)⇒b=1/2

From (5)

a=−(1/2)−c

Inserting values of a and b in (3)

0=−(−1/2−c)−3×(1/2)+2c
⇒3c=1
⇒c=1/3

Inserting value of c in (5)

a+(1/3)=−1/2
⇒a=−(1/2)−(1/3)
⇒a=−5/6

∴a=−5/6,b=1/2andc=1/3

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