A stone of mass 0.3 kg tied to the end of a string in a horizontal plane is whirled round in a circle of radius 1m with a speed of 40 rpm. What is the tension in the string? What is the maximum speed with which the stone can be whirled around if the string can withstand a maximum tension of 200N?

Solution

Given,
Mass of the stone, m = 0.3 kg
Length of the string, l = 1m
Angular velocity, $space straight omega equals 40 space rpm space equals space fraction numerator 4 straight pi over denominator 3 end fraction space rad divided by sec$
Therefore tension in the string,

$T equals m divided by omega squared equals 0.3 cross times 1 cross times open parentheses fraction numerator 4 straight pi over denominator 2 end fraction close parentheses squared equals 5.26 space straight N$
Let $straight omega subscript 0$ be the maximum angular speed of stone with which it can be whirled around.
The centrifugal force on the stone should not increase above maximum tension 200N. Otherwise the string will break.
∴ $m l straight omega subscript 0 superscript 2 space equals space 200$
$rightwards double arrow$ $0.3 cross times 1 cross times straight omega subscript 0 superscript 2 equals 200$
$rightwards double arrow$ $space space space space space space space space space space straight omega subscript 0 equals square root of fraction numerator 200 over denominator 0.3 end fraction end root space equals space 25.8 space rad divided by straight s$