A ball attached to the end of a string is swung around in a...

A ball attached to the end of a string is swung around in a circular path of radius $r$. If the radius is doubled and the linear speed is kept constant, the centripetal acceleration a) remains the same. d) increases by a factor of 4 . b) increases by a factor of 2 . e) decreases by a factor of 4 c) decreases by a factor of 2 .

A ball attached to the end of a string is swung around in a circular path of radius $r$. If the radius is doubled and the linear speed is kept constant, the centripetal acceleration
a) remains the same.
d) increases by a factor of 4 .
b) increases by a factor of 2 .
e) decreases by a factor of 4
c) decreases by a factor of 2 .

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Key Concepts

Uniform Circular Motion

Uniform circular motion refers to the movement of an object along a circular path at a constant speed. In this motion, although the speed remains constant, the velocity vector continuously changes direction, meaning the motion is accompanied by an acceleration directed towards the center of the circle.

Centripetal Acceleration

Centripetal acceleration is the inward acceleration required to keep an object moving in a circular path. It is defined by the formula a = v²/r, where v is the linear speed and r is the radius of the circle. This acceleration is responsible for changing the direction of the velocity vector, not its magnitude.

Dependence on Radius

In the context of circular motion, the centripetal acceleration is inversely proportional to the radius. When the radius is increased while keeping the linear speed constant (as per the formula a = v²/r), the centripetal acceleration decreases. Specifically, if the radius is doubled, the centripetal acceleration is reduced by a factor of 2.

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