Question

A mass $m$ attached to a string of length $l$ is suspended from the ceiling. Mass, $m$, is set into motion in a horizontal circle with string making an angle $\theta$ to the vertical. If T${}_m$ is the breaking tension of the string, the maximum radius of the circle is

• A. $\left(\frac{mg}{T_m}\right)l$
• B. $\left(\sqrt{1-\frac{m^2{g}^2}{T_m^2}}\right)l$
• C. $\frac{T_m}{mg}l$
• D. $\sqrt{\frac{mg}{T_m}l}$

Answer 1

Answer: (B)

$\left(\sqrt{1-\frac{m^2{g}^2}{T_m^2}}\right)l$

Let the tension in the string be $T$.
The forces acting on the mass are mg downward and T acting upward at angle $\theta$ with the vertical.
$T\cos \theta =mg$

$T\sin \theta =\frac{m{v}^2}{r}$

Since the max tension is $T_m,\cos \theta =\frac{mg}{T_m}$

$r=l\sin \theta =l\sqrt{1-{\cos }^2\theta }=l\sqrt{1-(\frac{mg}{T_m}{)}^2}$