Question

The moon is in a nearly circular orbit of radius r = 384000000 meters

and has a period of 27.32 days. What is the centripetal acceleration of
the moon toward the earth?

Answer 1

`2.72\cdot 10^(-3) m/s^2`

Step-by-step explanation:

The centripetal acceleration of an object in circular motion is the acceleration with which the object is attracted towards the center of the circular orbit. Mathematically, it is given by

`a=(v^2)/(r)`

where

v is the speed of the object

r is the radius of the orbit

The speed of the object is also given by the ratio between the circumference of the orbit and the orbital period, T:

`v=(2\pi r)/(T)`

Substituting into the previous equation, we find a new expression for the centripetal acceleration:

`a=(4\pi^2 r)/(T^2)`

In this problem:

- The radius of the orbit of the Moon is

`r = 384000000 m = 3.84\cdot 10^8 m`

- The period of the orbit is

`T=27.32 d \cdot 24\cdot 60\cdot 60 =2.36\cdot 10^6 s`

Therefore, the centripetal acceleration is:

`a=(4\pi^2 (3.84\cdot 10^8))/((2.36\cdot 10^6)^2)=2.72\cdot 10^(-3) m/s^2`