A simple pendulum of length $40 \mathrm{cm}$ oscillates with an angular amplitude of $0.04$ rad. Find the the linear amplitude of the bob

A ball is suspended by a thread of length $l$ at the point $O$ on an incline wall as shown. The inclination of the wall with the vertical is $\alpha .$ The thread is displaced through a small angle $\beta$ away from the vertical and the ball is released. Find the period of oscillation of pendulum. Consider both cases.

$\alpha >\beta$

A ball is suspended by a thread of length $l$ at the point $O$ on an incline wall as shown. The inclination of the wall with the vertical is $\alpha .$ The thread is displaced through a small angle $\beta$ away from the vertical and the ball is released. Find the period of oscillation of pendulum. Consider both cases.

$\alpha <\beta$

Figure shows two identical simple pendulums of length $l.$ One is tilled at an angle $\alpha$ and imparted an initial velocity $v_1$ toward mean position and at the same time other one is projected away from mean position at a velocity $v_2$ at an initial angular displacement $\beta .$ Find the phase difference in oscillations of these two pendulums.

i 

ii 

If the second pendulum bob is thrown at velocity $v_2$ at an angle $\beta$ from mean position, but on other side of mean position, find the phase difference in the two SHMs now as shown in the figure.

I 

II 

An $L$-shaped bar of mass $M$ is pivoted at one of its end so that it can freely rotate in a vertical plane, as shown in the figure.

Find the value of ${\theta }_0$ at equilibrium.

An $L$-shaped bar of mass $M$ is pivoted at one of its end so that it can freely rotate in a vertical plane, as shown in the figure.

If it is slightly displaced from its equilibrium position, find the frequency of oscillation.