Answer:
[tex]a_{cp}=\frac{2\pi^2 d}{T^2}[/tex]
Explanation:
The equation for centripetal acceleration is
[tex]a_{cp}=\frac{v^2}{r}[/tex]
Where v is the tangential velocity and r the radius of the orbit.
To calculate v we notice that the moon travels the circumference of the orbit [tex]C=2\pi r[/tex] in a time T, so we have
[tex]v=\frac{C}{T} =\frac{2\pi r}{T}[/tex]
Putting all together:
[tex]a_{cp}=\frac{v^2}{r}=\frac{(\frac{2\pi r}{T})^2}{r}=\frac{4\pi^2 r}{T^2}[/tex]
And since the radius r is d/2, where d is the diameter of the moon's orbit, we can write the formula asked:
[tex]a_{cp}=\frac{4\pi^2 r}{T^2}=\frac{4\pi^2 d}{2T^2}=\frac{2\pi^2 d}{T^2}[/tex]
