91Ó°ÊÓ

Two balls of equal mass $m$in string role="math" localid="1649866390610" $A$and $B$.

The radius of circular motion of two balls are$r$and$2r$

The free-body diagram shows that the downward gravitational force is balanced by an upward component of the tension, leaving the radial component of the tension to cause the centripetal acceleration. Newton’s second law is

$\underset{}{∑}{F}_r=T=\frac{m{v}^2}{r}$

The above tension equation is applied for the given cases of free body diagram of $FIGUREQ8.5$

case ($A$): The tension at mass with velocity and radius of circular motion gives

localid="1649854235133" $T_A=\frac{m{v}^2}{r}$

case ($B$): The tension at mass with velocity and radius of circular motion gives

localid="1649854244917" $T_B=\frac{m{v}^2}{2r}=0.5\frac{m{v}^2}{r}$

The Tensions of each string for different cases are ranked in the order$T_A>T_B$

Angular velocity is given by $v=r\mathrm{Ó\neg }$

case ($A$): The tension at mass with angular velocity and radius of circular motion gives

$T_A=\frac{m{v}^2}{r}=\frac{m{r}^2{\mathrm{Ï–}}^2}{r}=mr{\mathrm{Ó\neg }}^2$

case ($B$): The tension at mass with angular velocity and radius of circular motion gives

$T_B=\frac{m{v}^2}{2r}=\frac{m4r^2{\mathrm{Ï–}}^2}{2r}=2mr{\mathrm{Ó\neg }}^2$

The Tensions of each string for different cases are ranked in the order$T_B>T_A$