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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` using dimensional analysis.

Text Solution

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We have `" "[T] prop [G]^(a)[M]^(b)[R]^(c )`
`rArr [M]^(a)[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) rArr T prop sqrt((R^(3))/(GM))`
The actual expression is T `= 2pi sqrt((R^(3))/(GM))`
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