A disc placed on a frictionless horizontal plank is rotating with a constant angular velocity $\omega$ about its central vertical axis. Now the plank is made to move with a constant acceleration $\alpha$ on a straight path. If you assume initial location of the centre of disc as origin, direction of motion of the plank in negative $y$-axis of the coordinate system attached with the plank, which of the following equations represent trajectroy of instantaneous centre of rotation of the disc.

A rigid massless rod of length $3l$ has two masses attached at each end as shown in the figure. The rod is pivoted at point $P$ on the horizontal axis. When released from the initial horizontal position, its instantaneous angular acceleration will be

A uniform string of length $20 \mathrm{m}$ is suspended from a rigid support. A short wave pulse is introduced at its lowest end. It starts moving up the string. The time taken to reach the support is 

(Take, $\mathrm{g}=10\mathrm{ }\mathrm{m} {\mathrm{s}}^{-2}$)

Consider regular polygons with number of sides $n=\mathrm{3,4},5\dots$ as shown in the figure. The center of mass of all the polygons is at height $h$ from the ground. They roll on a horizontal surface about the leading vertex without slipping and sliding as depicted. The maximum increase in height of the locus of the center of mass for each polygon is $\mathrm{\Delta }$ . Then, $\mathrm{\Delta }$ depends on $n$ and $h$ as

A transverse wave propagating along $x$ -axis is represented by

$y\left(x,t\right)=8.0\sin \left(0.5\pi x-4\pi t-\frac{\pi }{4}\right)$

where $x$ is in meters and $t$ is in seconds. The speed of the wave is

A mass $m$ is supported by a massless string wound around a uniform hollow cylinder of mass m and radius R. If the string does not slip on the cylinder, with what acceleration will the mass fall on release?

A uniform rod of length ' ${\ell }^{\text{'}}$ 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) $\frac{m{\ell }^2}{12}{\omega }^2\sin \theta$ about the centre of mass (CM) to the torque provided by the horizontal and vertical forces $F_H$and $F_v$ about the CM. The value of $\theta$ is then such that:

The transverse displacement at position $x$ and time $t$ in a string due to a travelling wave is given by $y(x, t)=$$3.0\cos (\pi x-4\pi t)\mathrm{cm}$, where $x$ is in centimeters and $t$ is in seconds. Which of the following statements is wrong?