How will the centripetal acceleration of the body change if the radius of the circle doubles and the speed remains unchanged?

Given:

v1 = v2 = v – the linear speed of a body moving in a circle does not change;

r2 = 2 * r1 – the radius of the circle was doubled.

It is required to determine a2 / a1 – how the centripetal acceleration of the body will change with an increase in the radius of the circle.

The centripetal acceleration in the first case will be equal to:

a1 = v1 ^ 2 / r1 = v ^ 2 / r1.

The centripetal acceleration in the second case will be equal to:

a2 = v2 ^ 2 / r ^ 2 = v ^ 2 / (2 * r1).

Then:

a2 / a1 = (v ^ 2 / (2 * r1)) / (v ^ 2 / r1) = v ^ 2 * r1 / (v ^ 2 * 2 * r1) = 1/2, that is, will decrease by 2 times.

Answer: with an increase in the radius of the circle by 2 times, the centripetal acceleration will decrease by 2 times.