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A large fluid star oscillates in shape u...

A large fluid star oscillates in shape under the influence of its own gravitational field. Using dimensional analysis, find the expression for period of oscillation (T) in terms of radius of star (R ), mean density of fluid `(rho)` and universal gravitational constant (G).

Text Solution

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Let `T = KR^a rho^b G^c ….. 9i)`
`[M^0 L^0 T^1] = [L]^a [ML^(-3)]^b[M^(-1) L^3 T^(-2)]^c = M^(b-c) L^(a - 3b+ 3c) T^(-2c)`
Applying principle of homogeneity of dimensions, we get
`b -c =0 , a - 3b + 3c = 0 ,-2c =1, c = -(1)/(2)`
`:. b = c = (1)/(2) and a-3(-(1)/(2)) +3 (-(1)/(2)) = 0, a= 0`
Putting in (i), we get `T= KR^0 rho^(-1//2) G^(-1//2) = K(rhoG)^(-1//2)`
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