A point moves with a constant modulo speed along a circle of radius R. How will the centripetal acceleration of a point

A point moves with a constant modulo speed along a circle of radius R. How will the centripetal acceleration of a point change if its speed is doubled and the radius of the circle is halved?

Given:

v is the speed of a material point;

R is the radius of the circle along which the point is moving;

v1 = 2 * v – the speed of movement was increased by 2 times;

R1 = R / 2 – the radius of the circle was reduced by 2 times.

It is required to determine a1 / a – the change in the centripetal acceleration of the point.

In the first case, the centripetal acceleration will be equal to:

a = v ^ 2 / R.

In the second case, the centripetal acceleration will be equal to:

a1 = v1 ^ 2 / R1 = (2 * v) ^ 2 / (R / 2) = 4 * 2 * v ^ 2 / R = 8 * v ^ 2 / R.

Then:

a1 / a = (8 * v ^ 2 / R) / (v ^ 2 / R) = 8 times.

Answer: centripetal acceleration will increase by 8 times.