Moment of Inertia

From a triangular lamina of side a and mass $m$ , a small triangular Lamina made by joining mid-points of sides of triangular lamina is cut. If Moment of Inertia of complete Lamina about COM is $I_o$ and Moment of Inertia of remaining Lamina about centroid is $\frac{x{I}_0}{16}$, then find the value of $x$.

Four point masses $($each of mass $m$$)$ are arranged in the $X-Y$ plane. The moment of inertia of this array of masses about $Y$-axis is 

M.O.I is the ratio of 

A solid sphere, a thin ring and a cylinder of same mass and radius, held at rest are released simultaneously to roll down on an inclined plane. Which one will reach the bottom first?

Point masses $1, 2, 3$ and $4 \mathrm{kg}$ are lying at the points $(0, 0, 0), (2, 0, 0), (0, 3, 0)$ and $(-2, -2, 0)$, respectively. The moment of inertia of this system about the $x$-axis will be,

Four similar point masses (each of mass $m$) are placed on the circumference of a disk of mass $M$ and radius $R$. The moment of inertia of the system about the normal axis through the center $O$ will be,

On the marks of $20 \mathrm{cm}$ and $70 \mathrm{cm}$ of a light meter scale, two weights, $1 \mathrm{kg}$ and $4 \mathrm{kg}$, respectively, are placed. The moment of inertia (in $\mathrm{kg} {\mathrm{m}}^2$) about the vertical axis through the $100 \mathrm{cm}$ mark will be,

M.O.I is the ratio of 

M.O.I. of a disc is minimum about an axis 

The moment of inertia of hollow cylinder having density $\rho$ and height $h$ is $\frac{\pi \rho h{r}^4}{2}$.

Describe the term moment of inertia of hollow cylinder?

The moment of inertia of a hollow cylinder of mass $M$ and inner radius $R_1$ and outer radius $R_2$ about its central axis is

A flywheel is used in an automobile engine due to its:

Why fly wheels are used in vehicles?

The moment of inertia of a circular ring about an axis passing through its centre and perpendicular to its plane is $200 \mathrm{g} {\mathrm{cm}}^2$. If the radius of the ring is $5 \mathrm{cm}$, find the mass of the ring.

Fly wheel is an important part of an engine because:

The moment of inertia of a rigid body depends on:

A uniform thin rod $\mathrm{AB}$ of length $L$ and mass $m$ is undergoing rotation about a fixed axis passing through end $\mathrm{A}$ and perpendicular to the rod, such that end $\mathrm{A}$ remains stationary as shown. The kinetic energy of section $\mathrm{AP}$ of the rod is equal to the kinetic energy of section $\mathrm{BP}$ of the rod at an instant. Then what is the value of ${\left(\frac{\mathrm{A}\mathrm{B}}{\mathrm{AP}}\right)}^3$?

($\mathrm{AB}$ and $\mathrm{AP}$ are lengths of respective parts of the rod)

A solid cylinder is made of radius $R$ and height $3R$ having mass density $\rho$ . Now two half spheres of radius $R$ are removed from both ends. The moment of inertia of remaining portion about axis ${\mathrm{ZZ}}^{\text{'}}$ can be calculated as $\frac{29}{6K}\pi R^5\rho$ . Find $K$.

Two identical water bottles one empty and the other filled with water are allowed to roll down an inclined plane. Which one of them reaches the bottom first? Explain your answer.