Question

A ball attached at the end of a string revolving in a horizontal circle of radius,R,at whirle a constant speed,v.The tension in the string is F.What is the force needed to increase the speed by 30%?

Answer 1

Let the radius of the circle be $R$, the angle between the string and the vertical be $\theta$, the tension in the string be $T$, the speed of the ball is $v$. Let's apply the Newton's Second Law of Motion in directions $x$ and $y$:

$$Tcos\theta=\dfrac{mv^2}{r},$$ $$Tsin\theta=mg.$$

From the first equation we can find the tension in the string:

$$T=\dfrac{mv^2}{rcos\theta}.$$

If the initial speed of the ball increased by 30%, the final speed will be $v+0.3v=1.3v$. Then, we can write the force of tension needed to increase the speed by 30%:

$$T=\dfrac{m(1.3v)^2}{rcos\theta}=1.69\cdot\dfrac{mv^2}{rcos\theta}=1.69T.$$