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It is known that the time of revolution ...

It is known that the time of revolution T of a satellite 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, 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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