Question

An undiscovered planet, many lightyears from Earth, has one moon in a periodic orbit. This moon takes time ? on average to complete one nearly circular revolution around the unnamed planet. If the diameter of the moon's orbit is ?, calculate the moon's radial (or centripetal) acceleration a_c.

a_c=

Answer 1

Time taken by the moon to complete one revolution = t

Diameter of the moon's orbit = D

Radius of the moon's orbit = R = D/2

Orbital speed of the moon = V

The moon completes one revolution in time 't' therefore it covers a distance of '2$\pi$R' in time 't'.

Vt = 2$\pi$R

$V = \frac{2\pi R}{t}$

Centripetal acceleration of the moon = a_c

$a_{c} = \frac{V^{2}}{R}$

(27R/t)2 ac = ?

$a_{c} = \frac{4\pi^{2} R}{t^{2}}$

$a_{c} = \frac{4\pi^{2} (D/2)}{t^{2}}$

$a_{c} = \frac{2\pi^{2} D}{t^{2}}$

The moon's centripetal acceleration is given by,

$a_{c} = \frac{2\pi^{2} D}{t^{2}}$