Perpendicular Axis Theorem

Q.

Analogue of mass in rotational motion is:

1. Moment of inertia

2. Angular momentum

3. Gyration

4. None of these

Q.

Let 'I' be the moment of inertia of a uniform square plate about an axis AB that passes through its centre and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle $\theta$ with AB. The moment of inertia of the plate about the axis CD is then equal to

1. $I$
2. $I{\sin }^2\theta$
3. $I{\cos }^2\theta$
4. $I{\cos }^2\left(\frac{\theta }{2}\right)$

Q. Moment of inertia of a ring of radius ${}^{\prime }{r}^{\prime }\text{m}$ about an axis passing through the $COM$, perpendicular to its plane is $I{\text{kg-m}}^2$. The moment of inertia of the same ring about an axis passing through the diameter of the ring is $nI{\text{kg-m}}^2$. Find the value of $n$.

Q. 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 the figure. The moment of inertia of the loop about the axis XX' is

1. $\frac{\rho L^3}{8{\pi }^2}$
2. $\frac{\rho L^3}{16{\pi }^2}$
3. $\frac{5\rho L^3}{16{\pi }^2}$
4. $\frac{3\rho L^3}{8{\pi }^2}$

Q. Find the moment of inertia of ring of mass $M$ and radius $R$ about the axis passing through the ring diametrically. If moment of inertia about the axis passing through its center of mass and perpendicular to the plane is $M{R}^2$.

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

Q. $ABCD$ is a square plate with centre $O$. The moment of inertia of the plate about the axes $1,2,3$ and $4$ are $I_1,I_2,I_3\&I_4$​ respectively. It follows that

1. $I_1+I_2=I_3+I_4$
2. $I_1+I_4=I_2+I_3$
3. $I_1+I_3=I_2+I_4$
4. $I_1-I_3=I_2-I_4$

Q. The moment of inertia of a uniform disc about the perpendicular axis through its centre of mass is $20{\text{kg m}}^2$. Then its moment of inertia about an axis along its diameter and in the plane of the disc will be

1. $20{\text{kg m}}^2$
2. $5{\text{kg m}}^2$
3. $10{\text{kg m}}^2$
4. $40{\text{kg m}}^2$

Q.

The moment of inertia of a thin square plate ABCD of uniform thickness about an axis passing through the centre O and perpendicular to the plane of the plate can NOT be given by :


where, $I_1,I_2,I_3,I_4$ are the moments of inertia of the plates about axes 1, 2, 3, 4 respectively.

1. $I_1+I_2$
2. $I_3+I_4$
3. $I_1+I_3$
4. $I_1+I_2+I_3+I_4$

Q. The moment of inertia of a uniform disc about the perpendicular axis through its centre of mass is $20{\text{kg m}}^2$. Then its moment of inertia about an axis along its diameter and in the plane of the disc will be

1. $20{\text{kg m}}^2$
2. $5{\text{kg m}}^2$
3. $10{\text{kg m}}^2$
4. $40{\text{kg m}}^2$

Q. A lamina lies in $y-z$ plane having moment of inertia about $x$ axis $I_x=6M{R}^2$ and moment of inertia about $y-$ axis $I_y=3M{R}^2$. Then, the moment of inertia of the lamina about $z$ axis is

1. $9M{R}^2$
2. $3M{R}^2$
3. $\sqrt{27}M{R}^2$
4. $\sqrt{45}M{R}^2$

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

1. $3M{R}^2$
2. $\frac{3}{2}M{R}^2$
3. $5M{R}^2$
4. $\frac{7}{2}M{R}^2$

Q. Let 'l' be the moment of inertia of an uniform square plate about an axis AB that passes through its centre and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle $\theta$ with AB. The moment of inertia of the plate about the axis CD is then equal to

1. $I$
2. $I{\sin }^2\theta$
3. $I{\cos }^2\theta$
4. $I{\cos }^2\frac{\theta }{2}$

Q. For the body shown in the figure:


Statement $1:I_z=I_x+I_y$
Statement $2:I_x=I_z+I_y$ and $I_y=I_x+I_z$
where $x$, $y$ and $z$ are the axes of rotation of the body.
Choose the correct option.

1. $\text{Statement}2\text{is correct and statement}1\text{is incorrect.}$
2. $\text{Statement}1\text{is correct and statement}2\text{is incorrect.}$
3. $\text{Both statements are correct}$
4. $\text{None of these}$

Q. A two dimensional lamina of arbitrary shape is placed in $x-y$ plane as shown in the figure. $\text{MOI}$ of lamina about $x$ and $y$ axes are $I_x=6{\text{kg-m}}^2$ and $I_y=4{\text{kg-m}}^2$ respectively. If point of intersection $\left(O\right)$ of the axes does not lies on lamina, then the $\text{MOI}$ of lamina about $z$ axis i.e. $\left(I_z\right)$ will be

1. $6{\text{kg-m}}^2$
2. $4{\text{kg-m}}^2$
3. $8{\text{kg-m}}^2$
4. $10{\text{kg-m}}^2$

Q. Three identical sperical shells, each of mass m and radius r are placed as shown in figure. Consider an axis XX' which is touching to two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about XX' axis is:

1. $3m{r}^2$
2. $\frac{16}{5}m{r}^2$
3. $4m{r}^2$
4. $\frac{11}{5}m{r}^2$

Q. Four identical thin rods each of mass M and length l, from a squre frame. Moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is

1. $\frac{2}{3}M{l}^2$
2. $\frac{13}{3}M{l}^2$
3. $\frac{1}{3}M{l}^2$
4. $\frac{4}{3}M{l}^2$

Q.

Two uniform identical rods each of mass $M$ and length $l$ are joined to form a cross as shown in figure. Find the moment of inertia of the cross about a bisector as shown dotted in the figure.

1. $\frac{M{L}^2}{6}$

2. $\frac{M{L}^2}{24}$

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

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

Q. Let 'l' be the moment of inertia of an uniform square plate about an axis AB that passes through its centre and is parallel to two of its sides. CD is a line in the plane of the plate that passes through the centre of the plate and makes an angle $\theta$ with AB. The moment of inertia of the plate about the axis CD is then equal to

1. $I$
2. $I{\sin }^2\theta$
3. $I{\cos }^2\theta$
4. $I{\cos }^2\frac{\theta }{2}$

Q. Find the moment of inertia of uniform circular disc placed on the horizontal surface having origin as center. If the moment of inertia of disc along the axis passing through the the diameter is $2{\text{kg m}}^2$, find the moment of inertia of disc about the axis perpendicular to the plane of disc and passing through its centre.

1. $2{\text{kg m}}^2$
2. $4{\text{kg m}}^2$
3. $6{\text{kg m}}^2$
4. $8{\text{kg m}}^2$

Q. Four identical thin rods each of mass M and length l, from a squre frame. Moment of inertia of this frame about an axis through the centre of the square and perpendicular to its plane is

1. $\frac{2}{3}M{l}^2$
2. $\frac{13}{3}M{l}^2$
3. $\frac{1}{3}M{l}^2$
4. $\frac{4}{3}M{l}^2$

Q. A rectangular lamina of mass $20\text{kg}$ having sides $6\text{m}$ and $3\text{m}$ is placed on the horizontal surface. Find moment of inertia of the lamina about the axis passing through the center and perpendicular to its plane.

1. $75{\text{kg m}}^2$
2. $45{\text{kg m}}^2$
3. $15{\text{kg m}}^2$
4. $55{\text{kg m}}^2$

Q. For the given uniform square lamina $ABCD$ of side length $l$, whose centre is $\text{O}$, identify the correct option.

1. $\sqrt{2}{I}_{AC}=I_{EF}$
2. $I_{AC}=2I_{EF}$
3. $I_{AC}=I_{EF}$
4. $I_{AC}=\sqrt{2}{I}_{EF}$

Q. Find the $\text{MOI}$ of a uniform square plate of side length $a=10\text{m}$ and mass $5\text{kg}$ about an axis passing through its centre as shown in figure. $(\theta ={30}^{\circ })$.

1. $I_{AB}=\frac{125}{6}{\text{kg.m}}^2$
2. $I_{AB}=\frac{125\sqrt{3}}{2}{\text{kg.m}}^2$
3. $I_{AB}=\frac{125}{3}{\text{kg.m}}^2$
4. $I_{AB}=\frac{250}{3}{\text{kg.m}}^2$

Q. 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 the figure. The moment of inertia of the loop about the axis XX' is

1. $\frac{\rho L^3}{8{\pi }^2}$
2. $\frac{\rho L^3}{16{\pi }^2}$
3. $\frac{5\rho L^3}{16{\pi }^2}$
4. $\frac{3\rho L^3}{8{\pi }^2}$

Q. Three identical discs $(D_1,D_2,D_3)$ are arranged as shown in figure. The moment of inertia of the system about the axis $O{O}^{\prime }$ is given by $Nm{r}^2$. Then, the value of $N$ is

Q. Three identical discs $(D_1,D_2,D_3)$ are arranged as shown in figure. The moment of inertia of the system about the axis $O{O}^{\prime }$ is given by $Nm{r}^2$. Then, the value of $N$ is

Q. Moment of inertia of a ring of radius ${}^{\prime }{r}^{\prime }\text{m}$ about an axis passing through the $COM$, perpendicular to its plane is $I{\text{kg-m}}^2$. The moment of inertia of the same ring about an axis passing through the diameter of the ring is $nI{\text{kg-m}}^2$. Find the value of $n$.