Moment of Inertia of a Rod

Q. A wire of mass $M$ and length $L$ is bent in form of a square. Calculate the MI of the square about the central axis perpendicular to the plane.

1. $\frac{M{L}^2}{2}$
2. $\frac{M{L}^2}{24}$
3. $\frac{M{L}^2}{36}$
4. $\frac{M{L}^2}{48}$

Q. Moment of inertia of a uniform disc of internal radius $r$ and external radius $R$ and mass $M$ about an axis through its centre and perpendicular to its plane is

1. $\frac{1}{2}M(R^2-r^2)$
2. $\frac{1}{2}M(R^2+r^2)$
3. $\frac{M(R^4+r^4)}{2(R^2+r^2)}$
4. $\frac{1}{2}\frac{M(R^4+r^4)}{(R^2-r^2)}$

Q. Find the moment of inertia ($\text{MI}$) of an uniform rod of mass $3\text{kg}$ and length $2\text{m}$ about an axis ($y{y}^{\prime }$) perpendicular to the plane of rod and passing through it's one end ($A$) as shown in the figure.

1. ${\text{1 kg-m}}^2$
2. ${\text{3 kg-m}}^2$
3. ${\text{4 kg-m}}^2$
4. ${\text{12 kg-m}}^2$

Q. Three thin rods each of length $L$ and mass $M$ are placed along $x,y$ and $z$ axes in such a way that one end of each of the rods is at the origin.

The moment of inertia of this system about $z$-axis is

1. $\frac{2M{L}^2}{3}$
2. $\frac{4M{L}^2}{3}$
3. $\frac{5M{L}^2}{3}$
4. $\frac{M{L}^2}{3}$

Q. Two thin rod of mass $m$ and length $l$ are joined together to make a ${}^{\prime }{L}^{\prime }$ shaped structure. Find out moment of inertia of structure about an axis passing through their common joining point and perpendicular to the plane of structure.

1. $\frac{m{l}^2}{12}$
2. $\frac{m{l}^2}{3}$
3. $\frac{2m{l}^2}{3}$
4. $\frac{m{l}^2}{6}$

Q. A thin rod of length $4l$, mass $4m$ is bent at the points shown in the figure. What is the moment of inertia of the rod about the axis passing point $\text{O}$ and perpendicular to the plane of the paper?

1. $\frac{M{l}^2}{3}$
2. $\frac{10M{l}^2}{3}$
3. $\frac{M{l}^2}{12}$
4. $\frac{M{l}^2}{24}$

Q. The moment of inertia of a uniform rod of length $2l$ and mass $m$ about an axis $x{x}^{\prime }$ passing through its centre and inclined at an angle $\alpha$ is

1. $\frac{m{l}^2}{3}{\sin }^2\alpha$
2. $\frac{m{l}^2}{12}{\sin }^2\alpha$
3. $\frac{m{l}^2}{6}{\cos }^2\alpha$
4. $\frac{m{l}^2}{2}{\cos }^2\alpha$

Q. A thin uniform rod of mass $5\text{kg}$ is rotating about an axis perpendicular to it's plane, passing through midpoint of the rod of length $4\text{m}$. Find the moment of inertia of the rod about this axis.

1. $66.6{\text{kg.m}}^2$
2. $6.66{\text{kg.m}}^2$
3. $33.33{\text{kg.m}}^2$
4. $3.33{\text{kg.m}}^2$

Q. Four identical rods are joined end to end to form a square. The mass of each rod is $M$. The moment of inertia of the square about the central line $y{y}^{\prime }$ as shown in figure is

1. $\frac{M{l}^2}{3}$
2. $\frac{M{l}^2}{4}$
3. $\frac{M{l}^2}{6}$
4. none of these

Q. A uniform rod of length $L$ is free to rotate in a vertical plane about a fixed horizontal axis through $B$. The rod begins rotating from rest from its unstable equilibrium position. When it has turned through an angle $\theta$, its average angular velocity $\omega$ is given as:

1. $\sqrt{\frac{6g}{L}}\sin \theta$
2. $\sqrt{\frac{6g}{L}}\sin \frac{\theta }{2}$
3. $\sqrt{\frac{6g}{L}}\cos \frac{\theta }{2}$
4. $\sqrt{\frac{6g}{L}}\cos \theta$

Q. Two rods $\text{OA}$ and $\text{OB}$ of equal lengths and masses are lying in the $xy$ plane as shown in figure. Let $I_x,I_y$ and $I_z$ be the moments of inertia of both the rods about $x,y$ and $z$ axis respectively. Then,

1. $I_x>I_y$
2. $I_x=I_y$
3. $I_x<I_y$
4. $I_x+I_y=I_z$

Q. The mass density of the material of a $\text{T}$ joint is $2\text{kg/m}$. The dimensions of this joint are shown in the figure.


What is the approximate moment of inertia ($\text{MI}$) of the $\text{T}$ joint about an axis passing through their common joining point and perpendicular to the plane of structure?

1. $120{\text{kg-m}}^2$
2. $240{\text{kg-m}}^2$
3. $362{\text{kg-m}}^2$
4. $426{\text{kg.m}}^2$

Q. Two rods $OA$ and $OB$ of equal length and mass are lying on the $xy$ plane as shown in figure. Let $I_x,I_y$ and $I_z$ be the moment of inertia of the system about $x,y$ and $z$ axis respectively. Then,

1. $I_x=I_y>I_z$
2. $I_x=I_y<I_z$
3. $I_x>I_y>I_z$
4. $I_z>I_y>I_x$

Q. A uniform rod of length $L$ and mass $M$ is bent at the middle point $\text{O}$ as shown in figure. Consider an axis passing through middle point $\text{O}$ and perpendicular to the plane of the bent rod. The moment of inertia about this axis is

1. $\frac{1}{48}M{L}^2$
2. $\frac{1}{24}M{L}^2$
3. $\frac{1}{12}M{L}^2$
4. depends on $\theta$

Q. A thin uniform rod of mass $10\text{kg}$ is rotating about an axis passing through the end point of the rod of length $6\text{m}$ as shown in figure. Find the moment of inertia of the rod about this axis.

1. $24{\text{kg-m}}^2$
2. $43.2{\text{kg-m}}^2$
3. $12{\text{kg-m}}^2$
4. $38{\text{kg-m}}^2$

Q. A rod is of length 2 m and mass 1kg. What will be its moment of inertia about an axis perpendicular to its length and passing through one of its end?

1. $\frac{4}{3}kg-m^2$
2. $\frac{1}{3}kg-m^2$
3. $\frac{3}{4}kg-m^2$
4. $\frac{2}{3}kg-m^2$

Q. A rod is lying along x-axis with its left extreme at origin as shown in figure. The density of the rod increases by the rate, $\frac{dm}{dx}=5x\frac{kg}{m}$. The length of the rod is 2m. What will be its rotational inertia about Z axis?

1. $\frac{10}{3}kg-m^2$
2. $\frac{20}{3}kg-m^2$
3. $10kg-m^2$
4. $20kg-m^2$

Q. A thin uniform rod of mass $10\text{kg}$ and length $6\text{m}$ is rotating about an axis perpendicular to its plane, passing through the midpoint of the rod. Find the moment of inertia of the rod about this axis.

1. $120{\text{kg-m}}^2$
2. $30{\text{kg-m}}^2$
3. $60{\text{kg-m}}^2$
4. $360{\text{kg-m}}^2$

Q. Moment of inertia of a uniform rod $AB$ about axis $Y{Y}^{\prime }$ as shown in figure is

1. $\frac{m{L}^2{\sin }^2\alpha }{4}$
2. $\frac{m{L}^2{\cos }^2\alpha }{3}$
3. $\frac{m{L}^2{\sin }^2\alpha }{3}$
4. $\frac{m{L}^2{\cos }^2\alpha }{4}$

Q. Four thin rods of same mass $3\text{kg}$ and same length $2\text{m}$, form a square as shown in figure. Moment of inertia of this system about an axis passing through centre $\text{'O'}$ and perpendicular to its plane is $P{\text{kg-m}}^2$. The value of $P$ is

Q. A rod of length $l$ having uniformly distributed charge $Q$ is rotated about one end with constant frequency $f$. Its magnetic moment is

1. $\pi fQ{l}^2$
2. $\frac{2\pi fQ{l}^2}{3}$
3. $\frac{\pi fQ{l}^2}{3}$
4. $2\pi fQ{l}^2$

Q. A wire of length $16\text{m}$ and mass $8\text{kg}$ is bent in the form of a rectangle $\text{ABCD}$ with $\frac{\text{AB}}{\text{BC}}=2$. The moment of inertia of this wire frame about side $\text{BC}$ is

1. $88.49{\text{kg-m}}^2$
2. $108.5{\text{kg-m}}^2$
3. $50.5{\text{kg-m}}^2$
4. $38{\text{kg-m}}^2$

Q. A non-uniform rod $AB$ has a mass $M$ and length $2l$.The mass per unit length of the rod is $mx$ at a point of the rod distant $x$ from $A$. Find the moment of inertia of this rod about an axis perpendicular to the rod through $A$.

1. $2M{l}^2$
2. $4M{l}^2$
3. $6M{l}^2$
4. $8M{l}^2$

Q. Two rods $OA$ and $OB$ of equal length and mass are lying on the $xy$ plane as shown in figure. Let $I_x,I_y$ and $I_z$ be the moment of inertia of the system about $x,y$ and $z$ axis respectively. Then,

1. $I_x=I_y>I_z$
2. $I_x=I_y<I_z$
3. $I_x>I_y>I_z$
4. $I_z>I_y>I_x$

Q.

Find the moment of inertia of a uniform rod of mass $M$ and length $L$, about $OA$ {$PQ$ makes an angle of ${30}^{\circ }$ with $OA$ as shown in figure}.

1. $\frac{M{L}^2}{4}$

2. $\frac{M{L}^2}{12}$

3. $\frac{M{L}^2}{3}$

4. $\frac{M{L}^2}{8}$

Q. Four spheres of mass $M$ and radius $R$ each are touching each other as shown. Calculate the MI about the axis which passes through centre of the square and perpendicular to the plane.

1. $\frac{12}{5}M{R}^2$
2. $\frac{24}{5}M{R}^2$
3. $\frac{48}{5}M{R}^2$
4. $\frac{M{R}^2}{28}$

Q. Four thin rods of same mass $3\text{kg}$ and same length $2\text{m}$, form a square as shown in figure. Moment of inertia of this system about an axis passing through centre $\text{'O'}$ and perpendicular to its plane is $P{\text{kg-m}}^2$. The value of $P$ is

Q. The moment of inertia of a uniform rod of length $2l$ and mass $m$ about an axis $(X-X)$ passing through its centre and inclined at an angle $\alpha ={30}^{\circ }$ is

1. $\frac{m{l}^2}{12}$
2. $\frac{m{l}^2}{3}$
3. $\frac{m{l}^2}{6}$
4. $\frac{m{l}^2}{4}$