Question

use the general formulas for gravitational force and centripetal force to derive the relationship between speed and orbital radius for circular objects. The centripetal force is provided by gravity. set formulas equal to each other and solve for velocity

Answer 1

Answer:

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

Explanation:

The universal law of gravitation is defined as:

$F = G\frac{Mm}{r^{2}}$  (1)                

Where G is the gravitational constant, M and m are the masses of the two objects and r is the distance between them.

The centripetal force can be found by means of Newton's second law:

$F = ma$  (2)

Since it is a circular motion, the acceleration can be defined as:

$a = \frac{v^{2}}{r}$  (3)

Where v is the velocity and r is the orbital radius.

Replacing equation (3) in equation (2) it is gotten:

$F = m\frac{v^{2}}{r}$  (4)

Hence,

$m\frac{v^{2}}{r} = G\frac{Mm}{r^{2}}$

Then, v can be isolated:

$mv^{2} = G\frac{Mmr}{r^{2}}$

$mv^{2} = G\frac{Mm}{r}$

$v^{2} = G\frac{Mm}{mr}$

$v^{2} = \frac{GM}{r}$

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

So the relationship between speed and orbital radius is given by the expression $v = \sqrt{\frac{GM}{r}}$