Perpendicular Axis Theorem

Q. Theorem of perpendicular axes for moment of inertia is applicable to

1. Both (1) and (2)
2. Sphere
3. Hollow cylinder
4. Triangular lamina

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 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. From a disc of radius R and mass M, a circular hole of diameter R, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of disc about a perpendicular axis, passing through the centre?

1. $9M{R}^2/32$
2. $15M{R}^2/32$
3. $13M{R}^2/32$
4. $11M{R}^2/32$

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. (a) Prove the theorem of perpendicular axes. (Hint : Square of the distance of a point (x, y) in the x–y plane from an axis through the origin and perpendicular to the plane is x2+y2). (b) Prove the theorem of parallel axes. (Hint : If the centre of mass of a system of n particles is chosen to be the origin (∑ m _t r_t = 0)

Q.

The rod is released. What is the angular speed when it turns by ${180}^o$?

1. $\sqrt{\frac{46g}{7L}}$

2. $\sqrt{\frac{45g}{7L}}$

3. $\sqrt{\frac{47g}{7L}}$

4. $\sqrt{\frac{47g}{7L}}$

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. 

2. 

3. 

4. 

Q. (a) Find the moment of inertia of a sphere about a tangent to the sphere, given the moment of inertia of the sphere about any of its diameters to be 2MR2/5, where M is the mass of the sphere and R is the radius of the sphere. (b) Given the moment of inertia of a disc of mass M and radius R about any of its diameters to be MR2/4, find its moment of inertia about an axis normal to the disc and passing through a point on its edge.

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 is

1. $l_1+l_2$

2. $l_3+l_4$

3. $l_1+l_3$

4. $l_1+l_2+l_3+l_4$

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. 2. What are the limitations of dimensional formula?

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. Two identical rods each of mass 'M' and length 'l' are joined in crossed position as shown in figure. The moment of inertia of this system about a bisector would be

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

Q. Let $x$, $y$ and $z$ are the axes of rotation of the body, and satisfies all necessary condition for application of Perpendicular axis theorem i.e. $I_z=I_x+I_y$, then point of intersection of $x$, $y$ and $z$ axes

1. must coincide with $\text{COM}$ of the body.
2. must lie on the body.
3. may or may not be on the body but must lie on same plane.
4. None of these

Q. Find the MOI of a uniform square plate of mass $m$ and edge $a$ about one of its diagonals.

1. $\frac{M{a}^2}{6}$
2. $\frac{M{a}^2}{3}$
3. $\frac{M{a}^2}{9}$
4. $\frac{M{a}^2}{12}$

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

1. $\frac{M{L}^2}{12}$
2. $\frac{M{L}^2}{4}$
3. $\frac{M{L}^2}{6}$
4. $\frac{M{L}^2}{3}$

Q. Galileo's law of inertia is another name for newton's.............. law of motion.

1. third
2. first
3. second
4. none of these.

Q. Calculate the moment of inertia of a disc of mass $M$ and radius $R$ about an axis coinciding with diameter.

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

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. Four thin uniform rods, each of length $L$ and mass $M$ are joined to form a square. The moment of inertia of square about an axis along one of its diagonal is

1. $\frac{M{L}^2}{6}$
2. $\frac{4M{L}^2}{3}$
3. $\frac{3M{L}^2}{4}$
4. $\frac{2M{L}^2}{3}$

Q. Consider a uniform square plate fo side a and mass M. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

1. $\frac{5}{6}M{a}^2$
2. $\frac{1}{12}M{a}^2$
3. $\frac{7}{12}M{a}^2$
4. $\frac{2}{3}M{a}^2$

Q. A two dimensional object lies in $x-z$ plane having moment of inertia $I_X=M{R}^2$ and $I_Y=4M{R}^2$. Then, the moment of inertia of object along $z-$axis is

1. $I_Z=3M{R}^2$
2. $I_Z=5M{R}^2$
3. $I_Z=\sqrt{17}M{R}^2$
4. $I_Z=\sqrt{10}M{R}^2$