Question

Check the dimensional consistency of equation v is equal to root GM by R

Answer 1

$\maltese \: \red{\mid{\underline{\overline{\textbf{Answer}}}\mid}}$

Dimensional consistency of an equation refers to dimensional equality of both sides of equation.

Dimension of Left hand side should be equal to Dimension of Right hand side.

Now we have been given that equation is :

$\maltese \: v = \sqrt{ \frac{GM}{r} }$

This equation is the equation of Escape velocity for an object.

Now, dimension of R.H.S will be

$v\mapsto \: {l}^{1} {t}^{ - 1}$

Now on the right side of equation we have mass already in dimensional form and r is dimensionally L. The G can be represented dimensionally on right side.

$G = {m}^{ - 1} {l}^{3} {t}^{ - 2}$

After inputting the Value of G on R.H.S We get :

$\sqrt{ \frac{2GM}{r} } = \sqrt{ \frac{ {m}^{ - 1} {l}^{3} {t}^{ - 2} {m}^{1} }{l} }$

$\sqrt{ \frac{2GM}{r} } = \sqrt{ {l}^{2} {t}^{ - 2} } = {l}^{1} {t}^{ - 1}$

Therefore we get dimension on Both sides of equation are equal.

Therefore the equation $v = \sqrt{ \frac{2GM}{r} }$ is dimensionally consistent.

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BY SCIVIBHANSHU

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