Moment of Inertia & Radius of Gyration

At the corners of an equilateral triangle of side L, point masses, each of mass $2$ kg are placed. The moment of inertia of the system about an axis along any one side is (in kg$\text{-m}^{2}$)

The moment of inertia of two equal masses each of mass m at separation L connected by a rod of mass M about an axis passing through centre and perpendicular to the length of rod is

Let M and L be the mass and length of thin uniform rod respectively. In first case, axis of rotation is passing through centre and perpendicular to length of rod. In second case axis of rotation is passing through one end and perpendicular to length of rod. The ratio of radius of gyration in first case to second case is

Moment of inertia of a disc of mass $m$ and radius $R$ about the centre of mass and perpendicular to plane is $I_1$ and moment of inertia of disc at a distance $\frac{2R}{3}$ from centre and perpendicular to plane is $I_2$ then $I_1/I_2$ is

The moment of inertia of a solid sphere about an axis passing through centre of gravity is $\frac{2MR^2}{5}$ where M is the mass and R is the radius. The radius of gyration about a parallel axis at a distance $2R$ from first axis is

A solid sphere of radius R and mass M is rotating about its diameter. The moment of intertia of the solid sphere rotating about an axis at a distance $\frac{R}{3}$ from the centre and parallel to that diameter is

A Solid sphere has mass M and radius R. Its moment of inertia about a parallel axis passing through a point at a distance R/3 from its centre is

Moment of inertia of a disc of mass $M$ and radius 'R' about any of its diameter is $MR^2/4$. The moment of inertia of this disc about an axis normal to the disc and passing through a point on its edge will be, $(x/2)MR^2$. The value of $x$ is

Three rings each of mass $M$ and radius $R$ are arranged as shown in the figure. The moment of inertia of a system about axis $YY'$ will be

A square frame ABCD is formed by 4 identical rods each of mass $m$ and length $\ell$. This frame is in XY plane such that side AB coincides with x-axis and side AD along y-axis. The moment of inertia of the frame about x-axis is

Four solid spheres, each of diameter $\sqrt{5}$ cm and mass $0.5$ kg are placed with their centres at the corners of a square of side $4$ cm. The M.I. of the system about the diagonal of the square is $N \times 10^{-4}$ kg-m$2$, then $N$ is

Four point masses m, 2m, 3m and 4m are kept at the corners A, B, C and D respectively of a square ABCD of side '$b$'. The moment of inertia of the system about an axis perpendicular to the plane of the square and passing through the point D is

A system of five solid spheres each of mass '$m$' and radius '$r$' are rotating about an axis AA' as shown in figure. Hence the moment of inertia of the system about the axis of rotation AA' is

Two identical thin rods of mass $M\text{ kg}$ and length $L\text{ m}$ are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point $P$ and perpendicular to the plane of the rods is $\frac{x}{12}ML^2\text{ kg m}^2$. The value of $x$ is ______.

Radius of gyration of a thin uniform circular disc about the axis passing through its centre and perpendicular to its plane is $K_c$. Radius of gyration of the same disc about a diamter of the disc is $K_d$. the ratio $K_d : K_c$ is

The ratio of radius of gyration of circular disc about a tangential axis in its plane and of a circular ring of the same radius about a tangential axis in its plane is

A solid sphere of mass 5 kg and a disc of mass 4 kg have the same radius. The ratio of moment of inertia of the sphere about its tangent to the moment of inertia of the disc about a tangent in its plane will be $x : y$. The value of $x$ and $y$ respectively is

Two identical rings A and B of same mass and radius are revolving, ring A arounds its own diameter and ring B about tangential axis in its own plane. Both the rings A and B have same rotational kinetic energy. The ratio of the angular velocity of ring B to that of ring A is

The radius of gyration of a solid sphere of radius 'R' and mass 'M' about its diameter is $K_d$ and that about a tangent of a solid sphere is $K_t$. The ratio of $K_d$ to $K_t$ is

The radius of gyration K of a hollow sphere of mass M and radius R about an axis XY is equal to R as shown in figure. The distance of that axis from the center of the sphere is h. The value of h is

A solid sphere of mass M and a disc of mass $\frac{M}{2}$ have the same radius. The ratio of moment of inertia of the disc about a tangent in its plane to the moment of inertia of the sphere about its tangent will be

Solid sphere is rotating about an axis as shown in figure. If the radius of the sphere is 5 cm, then radius of gyration about that axis will be $\sqrt{x}$ cm. The value of x is

Moment of inertia about an axis AB for a rod of mass 40 kg and length 3 m is same as that of a solid sphere of mass of 10 kg and radius $R$ about an axis parallel to AB axis with separation of 3 m as shown in figure below. The value of $R$ is given as $\sqrt{\frac{\alpha}{2}}$. The value of $\alpha$ is ________.

A solid sphere of radius 10 cm is rotating about an axis which is at a distance 15 cm from its centre. The radius of gyration about this axis is $\sqrt{n}\text{ cm}$. The value of $n$ is

A thin wire of length $L$ and uniform linear mass density $\rho$ is bent into a circular loop with centre at '$O$' as shown in figure. The moment of inertia of the loop about the axis $XX'$ is

Moment of inertia of a thin uniform rod rotating about the perpendicular axis passing through its centre is I. If the same rod is bent in to a ring and its moment of inertia about its diameter is $I_1$, then the ratio $I/I_1$ is

Two loops $A$ and $B$ of radii $R_1$ and $R_2$ are made from uniform wire. If moment of inertia of loops $A$ and $B$ is $I_A$ and $I_B$ respectively then $\frac{R_1}{R_2}$ is (Given: $\frac{I_A}{I_B} = 27$)

A thin uniform rod of length $2$ m cross-sectional area 'A' and density 'd' is rotated about an axis passing through the centre and perpendicular to its length with angular velocity $\omega$. If the value of $\omega$ in terms of its rotational kinetic energy $E$ is $(\alpha E/Ad)^{1/2}$, then the value of $\alpha$ is

A thin wire of length ' $L$ ' and linear mass density ' $m$ ' is bent into a circular ring (in $x-y$ plane) with centre ' $C$ ' as shown in figure. The moment of inertia of the ring about an axis $yy'$ will be:

A circular disc has radius $R_1$ and thickness $T_1$. Another circular disc made of the same material has radius $R_2$ and thickness $T_2$. If the moment of inertia of both discs are same and $\frac{R_1}{R_2} = 2$ then $\frac{T_1}{T_2} = \frac{1}{\alpha}$. The value of $\alpha$ is ______.

A solid sphere $A$ of radius $R$ and mass $M$ is attached at a point to a smaller solid sphere $B$ of radius $r < R$ and mass $m < M$. Assume that the line joining their centres lies along the horizontal. The moment of inertia of the system calculated about a vertical axis passing through the centre of $A$ is $I_A$ and that about a vertical axis passing through the centre of $B$ is $I_B$. The difference $I_A - I_B$ is:

Two circular discs of radius each 10 cm are joined at their centres by a rod of length 30 cm and mass 600 gm as shown in figure.

If the mass of each disc is 600 gm and applied torque between two discs is $43 \times 10^{5}$ dyne.cm, the angular acceleration of the discs about the given axis $AB$ is $\underline{\qquad}\ \text{rad/s}^2$.

A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm. The moment of inertia of this pair of spheres about the tangent passing through the point of contact is $\underline{\qquad}\ \text{kg}\cdot\text{m}^2$.

A solid sphere of mass $M$ and radius $R$ is divided into two unequal parts. The smaller part having mass $M/8$ is converted into a sphere of radius $r$ and the larger part is converted into a circular disc of thickness $t$ and radius $2R$. If $I_1$ is moment of inertia of a sphere having radius $r$ about an axis through its centre and $I_2$ is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia $I_2/I_1 = \underline{\qquad}$

Given identical rings are arranged in a hexagonal plane pattern so as to touch each neighbouring ring as shown in figure. Each ring has mass M and radius R. The moment of inertia of the system of seven rings about an axis passing through the centre of central ring and normal to plane of all rings is

A uniform solid cylinder with radius R and length L has moment of inertia $I_1$ about the axis of the cylinder. A concentric solid cylinder of radius $R/2$ and length $L/2$ is carved out of the original cylinder. If $I_2$ is the moment of inertia of the carved out portion of the cylinder then $I_1/I_2$ is (Both $I_1$ and $I_2$ are about the axis of the cylinder)

The moment of inertia of a square loop made of four uniform solid cylinders, each having radius $R$ and length $L\,(R < L)$ about an axis passing through the mid points of opposite sides, is (Take the mass of the entire loop as $M$):

A uniform solid cylinder of length $L$ and radius $R$ has moment of inertia about its axis equal to $I_1$. A small co-centric cylinder of length $L/2$ and radius $R/3$ carved from this cylinder has moment of inertia about its axis equals to $I_2$. The ratio $I_1/I_2$ is ______.

Suppose there is a uniform circular disc of mass $M\text{ kg}$ and radius $r\text{ m}$ shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis $A$ of the disc is given by $\frac{x}{256}Mr^2$. The value of $x$ is ______.

Four particles each of mass M are placed at the corners of a square of side L. The radius of gyration of the system about an axis perpendicular to the square and passing through its centre is

Four particles each of mass M are placed at the corners of a square of side $L$. The radius of gyration of the system about an axis perpendicular to the square and passing through its centre is

A solid sphere and thin walled hollow sphere have same mass and same material. Which of them have greater moment of inertia about their diameter?

[ $I_h =$ moment of inertia of hollow sphere about an axis coinciding with its diameter, $I_s =$ moment of inertia of solid sphere about an axis coinciding with its diameter]

The moment of inertia of a thin uniform rod of mass ' $M$ ' and length ' $L$ ', about an axis perpendicular to length of the rod and at a distance ' $L/4$ ' from one end is

The moment of inertia of a solid sphere of mass ' $m$ ' and radius ' $R$ ' about its diametric axis is ' $I$ '. Its moment of inertia about a tangent in the plane is

The moment of inertia of a circular ring of mass M and diameter r about a tangential axis lying in the plane of the ring is:

Two spheres each of mass $M$ and radius $R$ are connected with a massless rod of length 4 R. The moment of inertia of the system about an axis passing through the centre of one of the spheres and perpendicular to the rod will be

$A, B$ and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure.

The moment of inertia of the given system about PQ axis is $\frac{x}{15}I$, where I is the moment of inertia of the disc about its diameter. The value of $x$ is ____________.

The moment of inertia of a solid disc rotating along its diameter is 2.5 times higher than the moment of inertia of a ring rotating in similar way. The moment of inertia of a solid sphere which has same radius as the disc and rotating in similar way, is $n$ times higher than the moment of inertia of the given ring. Here, $n =$ ________ Consider all the bodies have equal masses.

Moment of inertia of a solid sphere about its diameter is 'I'. It is then casted into 27 small spheres of same diameter. The moment of inertia of each small sphere about its diameter is

Moment of inertia of the rod about an axis passing through the centre and perpendicular to its length is ' $I_1$ '. The same rod is bent into a ring and its moment of inertia about the diameter is ' $I_2$ '. Then $I_1/I_2$ is

A thin metal wire of length ' L ' and mass ' M ' is bent to form semicircular ring as shown. The moment of inertia about $XX'$ is

Moment of inertia of a thin uniform rod rotating about the perpendicular axis passing through its centre is ' $I$ '. If the same rod is bent in the form of ring, its moment of inertia about the diameter is ' $I_1$ '. If $I_1 = xI$, then the value of ' x ' is

Two circular loops P and Q of radii ' r ' and ' nr ' are made respectively from a uniform wire. Moment of inertia of loop Q about its axis is 4 times that of loop $P$ about its axis. The value of $n$ is

Two discs A and B of same material and thickness have radii $R$ and $3R$ respectively. Their moments of inertia about their axis will be in the ratio

A rod of linear mass density 'λ' and length 'L' is bent to form a ring of radius 'R'. Moment of inertia of ring about any of its diameter is.

Moment of inertia of a rod of mass ' M ' and length ' L ' about an axis passing through its center and normal to its length is ' $\alpha$ '. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axis passing through its center and normal to plane containing cross is:

Two iron solid discs of negligible thickness have radii $R_1$ and $R_2$ and moment of inertia $I_1$ and $I_2$, respectively. For $R_2 = 2R_1$, the ratio of $I_1$ and $I_2$ would be $1/x$, where $x =$ _______.

A sphere of radius $R$ is cut from a larger solid sphere of radius $2R$ as shown in the figure. The ratio of the moment of inertia of the smaller sphere to that of the rest part of the sphere about the $Y$-axis is:

Let M and R be the mass and radius of a disc. A small disc of radius $R/3$ is removed from the bigger disc as shown in figure. The moment of inertia of remaining part of bigger disc about an axis $AB$ passing through the centre $O$ and perpendicular to the plane of disc is $\frac{4}{x}MR^2$. The value of $x$ is ____________.

Three spheres, each of mass ' $m$ ' and radius ' $r$ ' are placed as shown in figure. Consider an axis $YY'$, which is touching two spheres and passing through the diameter of third sphere. The moment of inertia of the system consisting of these three spheres about $YY'$ axis is

A uniform circular disc of radius ' $R$ ' and mass ' $M$ ' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius $R/2$ is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.

Three point masses, each of mass ' $m$ ' are placed at the corners of an equilateral triangle of side ' $L$ '. The moment of inertia of the system about an axis passing through one of the vertices and parallel to the side joining other two vertices will be

Four particles each of mass $1\text{ kg}$ are placed at four corners of a square of side $2\text{ m}$. Moment of inertia of system about an axis perpendicular to its plane and passing through one of its vertex is _____ $\text{kg m}^{2}$.

Ratio of radius of gyration of a circular disc to that of circular ring each of same mass and radius around their respective axes is

An annular ring has mass 10 kg and inner and outer radii are 5 m and 10 m respectively. Its moment of inertia about an axis passing through its centre and perpendicular to its plane is

The radius of gyration of a circular disc of radius R and mass m rotating about diameter as axis is

Moment of inertia of a disc about an axis passing through its centre and perpendicular to its plane is 'I'. The ratio of moment of inertia about a parallel axis tangential to its rim to that about a parallel axis passing through a point midway between the centre and the rim is

The moment of inertia of uniform circular disc is maximum about an axis perpendicular to the disc and passing through point

The moment of inertia of thin square plate PQRS of uniform thickness, about an axis passing through centre ' O ' and perpendicular to the plane of the plate is $(I_1, I_2, I_3, I_4$ are respectively the moments of inertia about axis $1, 2, 3, 4$ which are in the plane of the plate as shown in figure)

The moment of inertia of a thin rod about an axis passing through its mid point and perpendicular to the rod is $2400\text{ g cm}^{2}$. The length of the $400\text{ g}$ rod is nearly:

Three thin rods, each of mass ' $M$ ' and length ' $L$ ' are placed along $X, Y$ and Z axes which are mutually perpendicular. One end of each rod is at origin. M. I. of the system about Z axis is

Three thin rods, each mass ' 2 M ' and length ' L ' are placed along $x, y$ and z axis which are mutually perpendicular. One end of each rod is at origin. Moment of inertia of the system about x - axis is

Three balls of masses $2\text{ kg}, 4\text{ kg}$ and $6\text{ kg}$ respectively are arranged at centre of the edges of an equilateral triangle of side $2\text{ m}$. The moment of inertia of the system about an axis through the centroid and perpendicular to the plane of triangle, will be ________ $\text{kg m}^2$.

Two identical spheres each of mass $2\text{ kg}$ and radius $50\text{ cm}$ are fixed at the ends of a light rod so that the separation between the centers is $150\text{ cm}$. Then, moment of inertia of the system about an axis perpendicular to the rod and passing through its middle point is $\frac{x}{20}\text{ kg m}^2$, where the value of $x$ is ___________.

Two spheres of equal masses, one of which is a thin spherical shell and the other solid sphere, have the same moment of inertia about their respective diameters. The ratio of their radii is

The radius of gyration of a solid sphere of mass $5\text{ kg}$ about $XY$ is $5\text{ m}$ as shown in figure. The radius of the sphere is $\frac{5x}{\sqrt{7}}\text{ m}$, then the value of $x$ is:

A disc and a ring both have same mass and radius. The ratio of moment of inertia of the disc about its diameter to that of a ring about a tangent in its plane is

Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis $AB$ as shown in figure is $\sqrt{8/x}$. The value of $x$ is:

Two loops P and Q of radii $R_1$ and $R_2$ are made from uniform metal wire of same material. $I_p$ and $I_Q$ be the moment of inertia of loop P and Q respectively then ratio $R_1/R_2$ is $($ Given $I_P/I_Q = 27)$

A solid metallic sphere of radius ' $R$ ' having moment of inertia '$I$' about diameter is melted and recast into a solid disc of radius ' $r$ ' of a uniform thickness. The moment of inertia of a disc about an axis passing through its edge and perpendicular to its plane is also equal to '$I$'. The ratio $\frac{r}{R}$ is

A circular disc of radius ' $R$ ' and thickness $\frac{R}{6}$ has moment of inertia 'I' about an axis passing through its centre and perpendicular to its plane. It is melted and recasted into a solid sphere then moment of inertia of sphere about an axis passing through diameter is

Two solid spheres ( A and B) are made of metals having densities $\rho_A$ and $\rho_B$ respectively. If there masses are equal then ratio of their moments of inertia $\left(\frac{I_B}{I_A}\right)$ about their respective diameter is

Two loops ' $A$ ' and ' $B$ ' of radii ' $R_1$ ' and ' $R_2$ ' are made from uniform wire. If moment of inertia of ' A ' is ' $I_A$ ' and that of ' B ' is ' $I_B$ ', then $R_2/R_1$ is $\left[\frac{I_A}{I_B} = 27\right]$

Two circular loops P and Q of radii ' r ' and ' nr ' are made respectively from a uniform wire. Moment of inertia of loop Q about its axis is four times that of loop P about its axis. The value of ' $n$ ' is