Question

A steel ball is whirled on the end of a chain in a horizontal circle of radius R with a constant period T. If the radius of the circle is then reduced to 0.75R, while the period remains T, what happens to the centripetal acceleration of the ball?

Answer 1

When the radius of the circle is reduced while the period remains constant, the centripetal acceleration of the ball increases.

Step-by-step explanation:

When the radius of the circle is reduced to 0.75R while the period remains T, the centripetal acceleration of the ball increases.

Centripetal acceleration is given by the formula ac = v^2 / r, where v is the tangential velocity and r is the radius of the circle. Since the period T remains constant and the radius is reduced, the velocity of the ball must increase in order to cover the smaller circumference in the same amount of time, resulting in an increase in centripetal acceleration.

Answer 2

The centripetal acceleration will be increased to 1.33 of its initial state.

Step-by-step explanation:

Centripetal acceleration

The Centripetal acceleration of an object is the acceleration of the object along a circular path moving towards the center of the circular path. The centripetal acceleration is represented in the equation bellow

`a_(c) = (V^(2) )/(r)` ...................................... 1

where
`a_(c)` is the centripetal acceleration

v is the tangential velocity

and r is the radius.

How the Change of Radius Affects the Centripetal Acceleration

Reference to equation 1 the centripetal acceleration (
`a_(c)`) is inversely proportional (
`y = (k)/(x)`) to the radius of the circle or path. this means that when the radius increases the centripetal acceleration reduces and when the radius reduces the centripetal acceleration increases. The radius was reduced to 0.75R in the question that will amount to 1.33
`a_(c)` increase in the centripetal acceleration. This can be obtained by multiplying the centripetal acceleration by the inverse of 0.75 which is 1.33.

Therefore, when the radius is reduced by 0.75R , the centripetal acceleration of the steel ball will increase by 1.33
`a_(c)`. since the period is kept constant