Question

given that v is the speed, r is the radius and g is acceleration due to gravity. which combination of the v, r, and g will be dimensionless?

Answer 1

We know v = k garb where K is a dimensionless constant.

[v] = [garb]

[L T −1] = [(LT −2)a Lb]

If we equate the powers on both sides we get

  a+b = 1

-1 = -2a

On solving we get a= ½ and b = ½

 Therefore, v α (gr)1/2

 So v2/rg is dimensionless.

Answer 2

"The dimensionless combination of v, r, and g is given below.

Given:

$v = Speed \frac { m }{ s }$

$r = Radius (m)$

$g = Acceleration \frac { m }{ s^{ 2 } }$

Hence, to get the dimensionless relation, we need to arrange the given units in such a way that they can cancel each other.

Therefore, we can make the relation as,

$\frac { v^{ 2 }\left(\frac { m }{ s } \right)}{ r(m)\quad \times \quad g\left(\frac { m }{ s^{ 2}}\right)}$

It will be the expression which is dimensionless as the unit of $v^{ 2 }$ is cancelled with the product of units of R and g."