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It is known that the time of revolution ...

It is known that the time of revolution `T` of a satelite around the earth depends on the universal gravitational constant G, the mass of the earth M, and the radius of the circular orbit R. Obtain an expression for `T` using dimensional analysis.

Text Solution

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We have `[T]=[G]^(a)[M]^(a)[R]^(c)`
`[M]^(0)[L]^(0)[T]^(1)=[M]^(-a)[L]^(3a)[T]^(-2a)xx[M]^(b)xx[L]^(c)=[M]^(b-a)[L]^(c+3a)[T]^(-2a)`
Comparing the exponents
For `[T] : 1=-2a rArr a=-1/2`
For `[M] : 0=b-a rArr b=a=-1/2`
For `[L] : 0=c+3a rArr c=-3a=3/2`
Putting the values we get `T prop G^(-1//2) M^(-1//2) R^(3//2) prop sqrt(R^(3)/(GM))`
The actual expression is `T=2pisqrt(R^(3)/(GM))`
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