Question

Given Newton’s Law for the gravitational force: F = G Mm R2 and Newton’s Second Law Fnet = ma, find an expression for the Moon’s orbital speed. Hint - You will also need to use the definition of centripetal acceleration.

Answer 1

let us consider moon's obital speed is v

So gravitational force between earth and moon is equal to the moon's centripetal acceleration.

$\frac{GMm}{R^{2}}=\frac{mv^{2}}{R}$                           ................(EQ1)

where M is the mass of the earth,m is the mass of the moon and R is the radius of the circular orbit of the moon that is the distance between earth and moon.

So from EQ1 we get

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

Again from Newton's second law

$ma=\frac{GMm}{R^{2}}$

where a is gravitational acceleration at the moon. So

$a=\frac{GM}{R^{2}}$

Now put the value of a in EQ1

so $v=\sqrt{aR}$