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.
.

Options

A. `((m^2 gl^3 cos^4 theta)/(sin theta))^(1//2)`
B. `((m^2 gl^3 cos^4 theta)/(cos theta))^(1//2)`
C. `((m^2 gl^3 sin^3 theta)/(cos theta))^(1//2)`
D. `((m^2 gl^3 cos^3 theta)/(cos theta))^(1//2)`.

Answer

B

Explanation

(b) We start with the particle under a net force model in the `x` and `y` directions :
`sum F_x = ma_x : T sin theta = (mv^2)/( r)`
`sum F_y = ma_y : T cos theta =mg`
So `(sin theta)/(cos theta) = (v^2)/(rg)` and `v = sqrt(rg((sin theta)/(cos theta)))`
then `L = rmv sin 90^@ = rm sqrt(rg((sin theta)/(cos theta))) = sqrt(m^2 gr^3 (sin theta)/(cos theta))`
and since `r = l sin theta`,
`L = sqrt(m^2 gl^3 (sin^4 theta)/(cos theta))`.