$A$ small particle of mass $m$ is projected at an angle $\theta$ with the $x$-axis with an initial velocity $v_{0}$ in the $x-y$ plane as shown in the figure. For time $t < \frac{v_{0} \sin \theta}{g}$,the angular momentum of the particle is (where $\hat{i}, \hat{j}$ and $\hat{k}$ are unit vectors along the $x, y$ and $z$ axes respectively):

$A$ man is sitting in a smooth groove on a horizontal circular table at the edge by holding a rope joined to the centre. The moment of inertia of the table is $I$. The mass of the man is $M$. The man now pulls the rope so that he comes to the centre. The angular velocity of the table:

$A$ wheel of radius $0.5 \ m$ and a moment of inertia of $10 \ kg \cdot m^2$ is rotating freely at an angular speed of $70 \ rev/min$. The wheel can be stopped in $5.0 \ s$ by pressing a wet cloth against the rim and exerting a radially inward force of $88 \ N$. The coefficient of kinetic friction between the wheel and wet cloth is:

$A$ particle starts from the point $(0\,m, 8\,m)$ and moves with a uniform velocity of $3\, \hat{i} \,m/s$. After $5\,s$,the angular velocity of the particle about the origin will be:

Which one of the following four graphs best depicts the variation with $x$ of the moment of inertia $I$ of a uniform triangular lamina about an axis parallel to its base at a distance $x$ from it?

Consider a disc rotating in the horizontal plane with a constant angular speed $\omega$ about its centre $O$. The disc has a shaded region on one side of the diameter and an unshaded region on the other side as shown in the figure. When the disc is in the orientation as shown,two pebbles $P$ and $Q$ are simultaneously projected at an angle towards $R$. The velocity of projection is in the $y-z$ plane and is same for both pebbles with respect to the disc. Assume that $(i)$ they land back on the disc before the disc completes $\frac{1}{8}$ rotation,$(ii)$ their range is less than half disc radius,and $(iii)$ $\omega$ remains constant throughout. Then