A wheel starts from rest has an angular acceleration that is given by $\alpha =6t^2$. The angle it turns in time $t$ is given as:

A thin uniform rectangular plate of mass $2 \mathrm{kg}$ is placed in $X-Y$ plane as shown in the figure. The moment of inertia about $x$-axis is $I_x=0.2 \mathrm{kg} {\mathrm{m}}^2$ and the moment of inertia about $y$ - axis is $I_y=0.3 \mathrm{kg} {\mathrm{m}}^2$. The radius of gyration of the plate about the axis passing through $O$ and perpendicular to the plane of the plate is

An massless equilateral triangle EFG of side 'a' (As shown in figure) has three particles of mass $\mathrm{m}$ situated at its vertices. The moment of inertia of the system about the line EX perpendicular to EG in the plane of EFG is $\frac{\mathrm{N}}{20}{\mathrm{ma}}^2$ where $\mathrm{N}$ is an integer. The value of $\mathrm{N}$ is ___________ .

Four equal masses, $m$ each are placed at the corners of a square of length $\left(l\right)$ as shown in the figure. The moment of inertia of the system about an axis passing through $A$ and parallel to $DB$ would be :

A wheel of mass $10 \mathrm{kg}$ has moment of inertia equal to $0.1 \mathrm{kg} {\mathrm{m}}^2.$ Then its radius of gyration equals $\underline{ }$

$ABC$ is a plane lamina of the shape of an equilateral triangle. $D, E$ are mid-points of $AB, AC$ and $G$ is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through $G$ and perpendicular to the plane $ABC$ is $I_0$. If part $ADE$ is removed, the moment of inertia of the remaining part about the same axis is $\frac{N{I}_0}{16}$ where $N$ is an integer. Value of $N$ is:

Which one of the following four graphs best depict 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