A conical pendulum of length $l$ makes an angle $\theta = 45 ^ { \circ }$ with respect to $Z$-axis and moves in a circle in the $X Y$ plane. The radius of the circle is 0.4 m and its center is vertically below $O$. The speed of the pendulum, in its circular path, will be - (Take $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$)
(1) $0.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(2) $0.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(3) $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$
(4) $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

A mass $M$ hangs on a massless rod of length $l$ which rotates at a constant angular frequency. The mass $M$ moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity $\omega$. The angular momentum of $M$ about point $A$ is $L _ { A }$ which lies in the positive $z$ direction and the angular momentum of $M$ about $B$ is $L _ { B }$. The correct statement for this system is:
(1) $L _ { A }$ and $L _ { B }$ are both constant in magnitude and direction
(2) $L _ { B }$ is constant in direction with varying magnitude
(3) $L _ { B }$ is constant, both in magnitude and direction
(4) $L _ { A }$ is constant, both in magnitude and direction

A particle of mass $m$ is suspended from a ceiling through a string of length $L$. The particle moves in a horizontal circle of radius $r$ such that $r = \frac{L}{\sqrt{2}}$. The speed of particle will be:
(1) $\sqrt{rg}$
(2) $\sqrt{2rg}$
(3) $\sqrt{\frac{rg}{2}}$
(4) $2\sqrt{rg}$