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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. |
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Answer» We have `[T]=[G]^(@) [M]^(@) [R]^(@)` `[M]^(0)[L]^(0)[T]^(1)=[M]^(-a)[L]^(3a)[T]^(-2a)xx[M]^(b-a)[L]^(c+3a)[T]^(-2a)` Comparing the exponents For `[T]: 1=-2arArra=-(1)/(2)` For `[M]: 0-b-arArrb=a=-(1)/(2)` For `[L]: 0=c+3arArrc=-3a=(3)/(2)` Putting the values we get `Tprop G^(-1//2)M^(-1//2)R^(3//2)propsqrt((R^(3))/(GM))` The actual expression is `T=2pisqrt((R^(3))/(GM))` |
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