First slide

Angular momentum

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 $θ$ with the vertical. The magnitude of the angular mornentum of the bob about the vertical dashed line is

Moderate

A

${(\frac{m^{2} {gl}^{3} \cos^{4} θ}{\sin θ})}^{\frac{1}{2}}$
B

${(\frac{m^{2} {gl}^{3} \sin^{4} θ}{\cos θ})}^{\frac{1}{2}}$
C

${(\frac{m^{2} {gl}^{3} \sin^{3} θ}{\cos θ})}^{\frac{1}{2}}$
D

${(\frac{m^{2} {gl}^{3} \cos^{3} θ}{\sin θ})}^{\frac{1}{2}}$

Solution

We start with the particle under a net force model in the x and y directions:
$\sum_{} F_{x} = {ma}_{x} : T \sin θ = \frac{{mv}^{2}}{r}$
$\sum_{} F_{y} = {ma}_{y} : T \cos θ = mg$
So $\frac{\sin θ}{\cos θ} = \frac{v^{2}}{rg} and v = \sqrt{rg \frac{\sin θ}{\cos θ}}$
then $L = rmv \sin 90^{0} = rm \sqrt{rg \frac{\sin θ}{\cos θ}} = \sqrt{m^{2} {gr}^{3} \frac{\sin θ}{\cos θ}}$
and since r = l sin $θ$ ,
$L = \sqrt{m^{2} {gl}^{3} \frac{\sin^{4} θ}{\cos θ}}$

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