Question

In uniform circular motion, how does the acceleration change when the speed is increased by a factor of 3? When the radius is decreased by a factor of 2?

Short answer

The increase in velocity by a factor of three increases acceleration by a factor of 9 and the acceleration will increase by a factor of 2 when radius is decease by 2.

Step-by-step solution

Concept/Significance of circular motion

Circular motion is defined as the movement of an item around the perimeter of a circle or the rotation of an object around a circular route.

Determination of acceleration change when the speed is increased by a factor of 3.

The acceleration of the object in circular orbit is given by,

$a=\frac{v^2}{r}$

Here, v is the velocity of the object and r is the radius of circular orbit.

Substitute values in the above equation.

role="math" localid="1663738218966" $$\begin{gathered} a=\frac{{\left(3v\right)}^2}{r} \\ a=\frac{9v^2}{r} \end{gathered}$$

thus, the increase in velocity by a factor of three increases acceleration by a factor of 9.

Determination of the radius is decreased by a factor of 2.

The acceleration of the object in circular orbit is given by,

$a=\frac{v^2}{r}$

Here,v is the velocity of the object and r is the radius of circular orbit.

Substitute value of radius in the above equation.

$$\begin{gathered} a=\frac{\left(\right)v^2}{r/2} \\ a=\frac{2v^2}{r} \end{gathered}$$

The acceleration is inversely proportional to the radius,

Thus, the acceleration will increase by a factor of 2 when radius is decease by 2.