A ball is whirled on the end of a string in a horizontal circle of radius R at const v. By which one of the following means can the centripetal acceleration of the b increased by a factor of two? 13. Keep the radius fixed and increase the period by a factor of two. Keep the radius fixed and decrease the period by a factor of two. Keep the speed fixed and increase the radius by a factor of two. Keep the speed fixed and decrease the radius by a factor of two. A) B) C) D)

D

Explanation:

Centripetal acceleration is the square of velocity divided by radius:

a = v² / r

Velocity is equal to angular velocity times radius, so this can also be written as:

a = ω² r

And angular velocity is 2π divided by the period, so:

a = (2π / T) ² r

We can use these two equations to determine which scenario results in double the centripetal acceleration.

A) Keep the radius fixed and increase the period by a factor of two.

(2π / 2T) ² r

1/4 (2π / T) ² r

1/4 a

B) Keep the radius fixed and decrease the period by a factor of two.

(2π / ½T) ² r

4 (2π / T) ² r

4a

C) Keep the speed fixed and increase the radius by a factor of two.

v² / 2r

1/2 a

D) Keep the speed fixed and decrease the radius by a factor of two.

v² / ½r

2a

Only D doubles the centripetal acceleration.