An ant and a beetle are riding a spinning record on a record player. The ant stands on the outside edge of the record while the beetle stands on the record's label. The ant is four times as far from the center of rotation as the beetle. What is the ratio of the ant's centripetal acceleration to the beetle's?

Please explain :) Thank you!

Ac = w^2 R

w the same
R2 = 4 R1
so four times

To find the ratio of the ant's centripetal acceleration to the beetle's, we first need to understand the concept of centripetal acceleration and its relationship with the distance from the center of rotation.

Centripetal acceleration is the acceleration experienced by an object moving in a circular path. It is always directed towards the center of the circle. The formula for centripetal acceleration is:

a = (v^2) / r

Where:
- a is the centripetal acceleration
- v is the velocity of the object
- r is the radius of the circular path

In this case, we are given that the ant is four times as far from the center of rotation as the beetle. Let's say the beetle's distance from the center is r, then the ant's distance from the center is 4r.

If we assume that both the ant and the beetle have the same velocity, then we can compare their centripetal accelerations using the formula above.

For the beetle:
a_beetle = (v^2) / r

For the ant:
a_ant = (v^2) / (4r)

Now, to find the ratio of the ant's centripetal acceleration to the beetle's, we divide the ant's acceleration by the beetle's acceleration:

a_ant / a_beetle = [(v^2) / (4r)] / [(v^2) / r]

The v^2 terms cancel out, simplifying the ratio to:

a_ant / a_beetle = r / (4r) = 1/4

Therefore, the ratio of the ant's centripetal acceleration to the beetle's is 1/4.

Remember, in this analysis, we assumed that both the ant and the beetle have the same velocity. If this assumption is not mentioned in the problem, the ratio of their centripetal accelerations might be different.