A uniform rod of length 'l' is pivoted at one of its ends on a vertical shaft of negligible radius. When the shaft rotates at angular speed $\omega$  the rod makes an angle   $\theta$ with it (see figure). To find $\theta$ equate the rate of change of angular momentum (direction going into the paper)   $\frac{m{l}^2}{12}{\omega }^2\mathrm{s}\mathrm{i}\mathrm{n}?\theta \mathrm{c}\mathrm{o}\mathrm{s}?\theta$ about the centre of mass (CM) to the torque provided by the horizontal and vertical forces   ${\mathrm{F}}_{\mathrm{H}}$ and   ${\mathrm{F}}_{\mathrm{V}}$ about the $\mathrm{C}\mathrm{M}$   . The value of $\theta$ is then such that:

 

Two SHMs in same direction are represented by y1 = asinωt and y2 = asin(ωt + δ). When they superimpose, the initial phase difference of the resultant SHM is

The potential energy of a particle of mass 2 kg in motion along the x-axis is given by U = 2 (1 – cos 2x) J, where x is in meters. The period of small oscillations (in s) is

A child is standing with folded hands at the centre of a platform rotating about its central axis. The kinetic energy of the system is K. The child now stretches his arms so that the moment of inertia of the system becomes 3/2 of the initial moment of inertia. The kinetic energy of the system is now

If the absolute temperature of hot black body is raised by 2%, by how much percentage rate of heat energy radiated would be increased nearly?

Two moles of helium gas are taken over the cycle ABCDA as shown below in P-T diagram. Assuming the gas to be ideal, the work done on the gas in taking from A to B is