One end of a string is attached to a ball, with the other end held by a student such that the ball is swung in a horizontal circular path of radius R at a constant tangential speed. At a later time, the tension force exerted on the ball remains constant, but the length of the string is decreased to R4. What is the new tangential speed of the ball?

a. four times the original speed
b. two times the original speed
c. Half the original speed
d. One-fourth the original speed

Any object moves at circle path will exhibit tangential speed as the change in the direction of motion leads to the speed acting as tangent to the circular path.

So the tangential speed defines as distance covered by any object moving in a circular path within the given time interval. As the object will be covering the distance equals to circumference of circle path, then

[tex]\text { Tangential speed }=\frac{\text { Circumference of the circular path}}{\text { Time taken }}[/tex]

So if we consider the length of the string as the radius of the circular path, then the original speed of the ball will be

[tex]\text { Original Tangential speed }=\frac{2 \times \pi \times R}{T}=V[/tex]

If the radius is decreased to [tex]\frac{R}{4}[/tex], then the new tangential speed will be

[tex]\text { New Tangential speed }=\frac{2 \times \pi \times R}{T \times 4}=\frac{\pi \times R}{2 \times T}=\frac{V}{4}[/tex]

So, the new tangential speed will be one fourth of the original speed.