Question

Use the general formulas for gravitational force and centripetal force to derive the relationship between speed (v) and orbital radius (r) for circular orbits. Show the relationship you derived to your instructor before going on.

Answer 1

Solution :

We know that :

Formula for Gravitational force is given by :

$F_g=\frac{Gmn}{r^2}$

where, G is the gravitational constant

            M is the mass of the bigger body

            m is the mass of the smaller body

            r  is the distance between the two bodies.

And the formula for the centripetal force is given by :

$F_c=\frac{mv^2}{r}$

where, m is the mass of the rotating body

            v is the velocity

             r is the radius of rotation of the body.

We know that mathematically, the gravitational force is equal to the centripetal force of the body.

Therefore,

$F_g=F_c$

$\frac{GMm}{r^2}=\frac{mv^2}{r}$

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

Hence derived.