Moment of Inertia

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:

The ratio of radius of gyration of a solid sphere of mass $M$ and radius $R$ about its own axis to the radius of gyration of the thin hollow sphere of same mass and radius about its axis is :

A solid sphere and a solid cylinder of same mass and radius are rolling on a horizontal surface without slipping. The ratio of their radius of gyrations respectively $\left(k_{sph}:k_{cyl}\right)$ is $2:\sqrt{x}$. The value of $x$ is _______.

A ring and a solid sphere rotating about an axis passing through their centres have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is $\sqrt{\frac{2}{x}}$. The value of $x$ is _____.

A pulley of radius $1.5 \mathrm{m}$ is rotated about its axis by a force $F=12t-3t^2 \mathrm{N}$ applied tangentially (while $t$ is measured in seconds). If moment of inertia of the pulley about its axis of rotation is $4.5 \mathrm{kg} {\mathrm{m}}^2$, the number of rotations made by the pulley before its direction of motion is reversed, will be$\frac{K}{\pi }$. The value of $K$ is

A solid sphere and a ring have equal masses and equal radius of gyration. If sphere is rotating about its diameter and ring about an axis passing through the centre and perpendicular to its plane, then the ratio of radius is $\sqrt{\frac{x}{2}}$, then find the value of $x$.

Find the MOI of a system of two particles having the same mass m separated by a distance $d$ about the Centre$O$.

Three co-planar uniform identical discs, each of mass $M$ and radius $R$, are joined as shown in figure. The moment of inertia about indicated axis is

A rectangular loop has mass $M$ and sides $a$ and $b$. An axis OO' passes through the centre $C$ of the loop and is parallel to side $a$ (lie in the plane of the loop). Then the radius of gyration of the loop for the axis OO' is

A rectangular loop has mass $M$ and sides $a$ and $b$. An axis $O{O}^{\text{'}}$ passes through the centre $C$ of the loop and is parallel to side $a$ (lie in the plane of the loop). Then the radius of gyration of the loop, for the axis $O{O}^{\text{'}}$ is

How to calculate the moment of inertia from any axis?

The equatorial diameter of the earth is greater than the polar diameter. How would the moment of inertia of the earth change if some mass from near the equator were removed and transferred to the polar regions to make the earth a perfect  sphere ?

Define radius of gyration. What is the purpose of defining it?

What is the physical significance of moment of inertia ?

Define moment of inertia. Name the factors on which the moment of inertia depend.

For calculating the moment of inertia of a body can we assume the whole mass of the body to be concentrated at its centre of mass ?

Derive an expression for the Moment of inertia of a sphere about its diameter.

Derive an expression for the M.I. of a disc (ii) about its diameter.

Derive an expression for the M.I. of a disc (i )about an axis passing through its centre and perpendicular to its plane.

Derive an expression per the Moment of Inertia of a rectangular rod about an axis passing through its center of mass and perpendicular to its length.