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The period of revolution (T) of a planet...

The period of revolution (T) of a planet around the sun depends upon (i) radius (r ) of obit (ii) mass M of the sun and (iii) gravitational constant G. Prove that `T^2 prop r^3`

Text Solution

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Let `T=Kr^(a)M^(b)G^(c )" ""……"(i)`
where K is a dimensionless constant
Taking dimensions on both sides of equation (i), we get
`[M^(0)L^(0)T^(1)]= [L]^(a)[M]^(b)[M^(-1)L^(3)T^(-2)]^(c )`
Comparing the exponents of M, L and T on both sides, we get
`b-c=0, a+3c=0, -2c=1`
`implies c= (-1)/(2)`
`b=c=(-1)/(2)`
`a= -3c= -3xx((-1)/(2))=(3)/(2)`
`T=Kr^(3/2)M^(-1/2)G^(-1/2)`
or `T^(2)= (K^(2)r^(3))/(MG)`
`implies T^(2) propto r^(3)`
Hence proved.
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