Question

A conical pendulum consists of a bob of mass $m$ in motion in a circular path in a horizontal plane as shown in figure. During the motion, the supporting wire of length $l$, maintains a constant angle $\theta$ with the vertical. The magnitude of the angular momentum of the bob about the vertical dashed line is

• A. ${\left(\frac{m^{2}g{l}^3{\sin }^3\theta }{\cos \theta }\right)}^{\frac{1}{2}}$
• B. ${\left(\frac{m^{2}g{l}^3{\cos }^3\theta }{\sin \theta }\right)}^{\frac{1}{2}}$
• C. ${\left(\frac{m^{2}g{l}^3{\sin }^4\theta }{\cos \theta }\right)}^{\frac{1}{2}}$
• D. ${\left(\frac{m^{2}g{l}^3{\cos }^4\theta }{\sin \theta }\right)}^{\frac{1}{2}}$

Answer 1

The correct option is C ${\left(\frac{m^{2}g{l}^3{\sin }^4\theta }{\cos \theta }\right)}^{\frac{1}{2}}$

Formula used: $T=2\pi \sqrt{\frac{l\cos \theta }{g}},\omega =\frac{2\pi }{T},L=mvr$

Speed of bob $v=\omega r=\sqrt{\frac{g}{l\cos \theta }}r$

Radius of circle in which bob moves is
$r=l\sin \theta$

$\therefore L=mvr$

$L=m\left(\sqrt{\frac{g}{l\cos \theta }}l\sin \theta \right)\left(l\sin \theta \right)$

$\sqrt{\frac{m^{2}g{l}^3{\sin }^4\theta }{\cos \theta }}$
Final answer (b)