Centripetal force on a string

Problem

A ball of mass $m = 0{,}30\ \mathrm{kg}$ rotates on a horizontal circle of radius $r = 0{,}50\ \mathrm{m}$ with $v = 4{,}0\ \mathrm{m/s}$. Calculate the centripetal force the string must exert.

The string pulls the ball towards the centre: the tension provides the centripetal force $\vec F_c$.

Solution

Setup. In uniform circular motion the resultant force is centripetal, directed towards the centre, and equals $F_c = ma_c$ with $a_c = v^2/r$. Here the only horizontal force is the string tension, which therefore coincides with the centripetal force: $F_c = m\,a_c = \frac{mv^2}{r}$

Numerical substitution. $F_c = \frac{0{,}30 \cdot (4{,}0)^2}{0{,}50} = \frac{0{,}30 \cdot 16}{0{,}50} = \frac{4{,}8}{0{,}50} = 9{,}6\ \mathrm{N}$

$\ev{F_c = \frac{mv^2}{r} = 9{,}6\ \mathrm{N},\ \text{directed towards the centre}}$

This is the tension with which the string pulls the ball towards the centre. If the string couldn’t withstand $9{,}6$ N it would snap and the ball would fly off along the tangent.

Linked items

Topics: Dinamica Concepts: Forza centripeta · Accelerazione centripeta · Tensione Objects: Fune ideale