Question

A ball is whirled on the end of a string in a horizontal circle of radius R at constant speed v. Which way(s) can increase the centripetal acceleration of the ball by a factor of 9? Group of answer choices Keeping the speed fixed and decreasing the radius by a factor of 9 Keeping the radius fixed and increasing the speed by a factor of 3 decreasing both the radius and the speed by a factor of 9 Keeping the radius fixed and increasing the speed by a factor of 9 increasing both the radius and the speed by a factor of 9 Keeping the speed fixed and increasing the radius by a factor of 9

Answer 1

Explanation:

When an object moves in a circular path, due to the change in its velocity, the object possess centripetal acceleration. The formula for the centripetal acceleration is given by :

$a=\dfrac{v^2}{R}$

Where

v is the speed of the object

R is the radius of the circle path

Option (1) : Keeping the speed fixed and decreasing the radius by a factor of 9

$a=\dfrac{v^2}{R/9}$

$a=\dfrac{9v^2}{R}$

The centripetal acceleration of the ball by a factor of 9.

Option (2) : Keeping the radius fixed and increasing the speed by a factor of 3

$a=\dfrac{(3v)^2}{R}$

$a=\dfrac{9v^2}{R}$

The centripetal acceleration increases by a factor of 9.

Option (3) : Decreasing both the radius and the speed by a factor of 9.

$a=\dfrac{(v/9)^2}{R/9}$

$a=\dfrac{(v)^2}{9R}$

The centripetal acceleration decreases by a factor of 9.

Option (4) : Keeping the radius fixed and increasing the speed by a factor of 9

$a=\dfrac{(3v)^2}{R}$

$a=\dfrac{9v^2}{R}$

The centripetal acceleration increases by a factor of 9.

Option (5) : Increasing both the radius and the speed by a factor of 9

$a=\dfrac{(9v)^2}{9R}$

$a=\dfrac{9v^2}{R}$

The centripetal acceleration increases by a factor of 9.

Option (6) : Keeping the speed fixed and increasing the radius by a factor of 9

$a=\dfrac{(v)^2}{9R}$

$a=\dfrac{9v^2}{R}$

The centripetal acceleration increases by a factor of 9.

So, as the radius of the circle decreases, its centripetal acceleration increase. Also, if the speed of the object increases, its centripetal acceleration increase. Hence, this is the required solution.