Question

The orbital velocity 'v' of a satellite may depend on its mass 'm', and distance 'r', from the center of the earth and acceleration due to gravity 'g'. Obtain an expression for or ital velocity.

Answer 1

Explanation:

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Answer 2

Answer:

$v = \sqrt{rg}$

Explanation:

We know that velocity is directly proportional to mass, distance as well as acceleration due to gravity.

Hence, it can be written as,

$v$ ∝ $m$

$v$ ∝ $r$

$v$ ∝ $g$

Let us put a, b, and c respectively as their powers.

$v = (m)^{a} \ (r)^{b} \ (g)^{c}$

Substituting their dimensional values we get,

$LT^{-1} = [M]^{a} \ [L]^{b} \ [L]^{c} \ [T]^{-2c$

$LT^{-1}= [M^{a} \ L^{b+c} \ T^{-2c}]$

On comparing the two equations we get,

$a = 0\\-2c = -1$

∴ $c = \frac{1}{2}$

Now, $b +c =1$

$b + \frac{1}{2} = 1$

∴ $b = \frac{1}{2}$

Hence, the dimensional formula becomes,

$v = [m]^{0} \ [r]^{\frac{1}{2} } \ [g]^{\frac{1}{2} }$

$v = k \sqrt{rg}$

Let, $k=1$

$v = \sqrt{rg}$

Therefore, the expression for orbital velocity is $v = \sqrt{rg}$

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