Chapter 10, Problem 21A

A ball is swung in a horizontal circle as shown below. The ball swings from various lengths of rope at the speeds indicated. Rank the tension m the ropes from greatest to least.

To determine

The rank of the tension in the ropes from greatest to least.

Answer to Problem 21A

  $T_{\text{A}}>T_{\text{C}}>T_{\text{B}}$

Explanation of Solution

Introduction:

When a mass $m$ tied along a string moves with the speed $v$ in a circle of radius $r$ , then the tension in the string will be balanced by the centripetal force and is given by,

  $T=\frac{m{v}^2}{r}$

Using above formula, the tension in the string in case A will be,

  $\begin{array}{c}{T}_{\text{A}}=\frac{m{v}^2}{r}\\ =\frac{m{\left(3\right)}^2}{0.4}\\ =22.5m\end{array}$

Similarly, the tension in the string in case B will be,

  $\begin{array}{c}{T}_{\text{B}}=\frac{m{v}^2}{r}\\ =\frac{m{\left(3.2\right)}^2}{0.6}\\ =17.06m\end{array}$

And, the tension in the string in case C will be,

  $\begin{array}{c}{T}_{\text{C}}=\frac{m{v}^2}{r}\\ =\frac{m{\left(4.0\right)}^2}{0.8}\\ =20m\end{array}$

Conclusion:

Hence, the tension in the string is directly proportional to the square of velocity and inversely proportional to the radius.