Question

Check the dimensional consistency of orbital velocity v = sq root of GR

Answer 1

For dimensional consistency dimension of left side of equation must be same as the dimension of the right side of the equation

So for the left side the dimension is of velocity

so it is

v = L^1T^{-1}

now for the dimension of gravitational constant G is

G = M^{-1}L^3T^{-2}

Now the dimension of right side is given as

\sqrt{\frac{2GM}{R}} = \sqrt{\frac{M^{-1}L^3T^{-2}*M^1}{L}}

\sqrt{\frac{2GM}{R}} = \sqrt{L^2T^{-2}} = L^1T^{-1}

so here dimension of left side of equation is same as right side

so here dimensional consistency hold good

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