Perpendicular Axis Theorem

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.

The moment of inertia of a disc of mass M and radius R about an axis, which is tangential to the circumference of disc and parallel to its diameter, is

1. $\frac{3}{2}m{R}^2$

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

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

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

Q. Four point masses, each of value m, are placed at the corners of square ABCD of side l. The moment of inertia of this system about an axis passing through A and parallel to BD is

1. 3$m{l}^2$
2. $m{l}^2$
3. 2$m{l}^2$
4. $\sqrt{3}m{l}^2$

Q. Moment of inertia of a uniform rod of length L and mass M, about an axis passing through $\frac{L}{4}$ from one end and perpendicular to its length is:

1. $\frac{7}{36}M{L}^2$
   
2. $\frac{7}{48}M{L}^2$
   
3. $\frac{1}{48}M{L}^2$
   
4. $\frac{M{L}^2}{12}$

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. Three 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 rod is at the origin. The moment of inertia of this system about Z axis is

1. 
2. 
3. 
4. 

Q. For the given uniform square lamina ABCD, whose centre is O,

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

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. 
2. 
3. 
4. 

Q.

The moment of inertia of a circular ring is $I$ about an axis perpendicular to its plane and passing through its center. About an axis passing through the tangent of the ring in its plane, its moment of inertia is

Q. $\text{ABC}$ is a plane lamina of the shape of an equilateral triangle. $\text{D, E}$ are mid-points of $\text{AB, AC}$ and $\text{G}$ is the centroid of the lamina. The moment of inertia of the lamina about an axis passing through $\text{G}$ and perpendicular to the plane $\text{ABC}$ is $I_0$. If the $\text{ADE}$ is removed. The moment of inertia of the remaining part about the same axis is $\frac{N{I}_0}{16}$ where $N$ is an integer. The value of $N$ is

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. 
2. 
3. 
4. 

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

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. 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. 
2. 
3. 
4. 

Q. A thin square plate $ABCD$ of uniform thickness is shown in the figure. $I_1,I_2,I_3$ and $I_4$ are respectively the moments of inertia about axis $1,2,3$ and $4$ which are in the plane of the square plate. Then, the moment of inertia $I_0$ of the plate about an axis passing through its centre $\text{O}$ and perpendicular to the plane of the plate will be equal to : (Choose the correct option(s))

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 moments of inertia of two freely rotating bodies $A$ and $B$ are $I_A$ and $I_B$​ respectively( $I_A>I_B$​) and their angular momenta are equal. If $K_A$​ and $K_B$​ are their kinetic energies, then

1. $K_A=K_B$

2. $K_A\ne K_B$

3. $K_A<K_B$

4. $K_A=2K_B$

Q. Find the moment of inertia of a uniform rectangular plate of mass $M$ and edges of length $l$ and $b$ about its axis passing through the centre and perpendicular to it

1. $\frac{M{l}^2}{12}$
2. $\frac{M{b}^2}{12}$
3. $\frac{M(l+b{)}^2}{12}$
4. $\frac{M(l^2+b^2)}{12}$

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. 
2. 
3. 
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 thin square plate $ABCD$ of uniform thickness is shown in the figure. $I_1,I_2,I_3$ and $I_4$ are respectively the moments of inertia about axis $1,2,3$ and $4$ which are in the plane of the square plate. Then, the moment of inertia $I_0$ of the plate about an axis passing through its centre $\text{O}$ and perpendicular to the plane of the plate will be equal to : (Choose the correct option(s))

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. Perpendicular axis theorem is applicable only to

1. two dimensional objects.
2. one dimensional objects.
3. All of the above
4. three dimensional objects.

Q. A symmetric lamina of mass $M$ consists of a square shape with a semicircular section over each of the edge of the square as shown in the figure. The side of the square is $2a.$ The moment of inertia of the lamina about an axis through its center of mass and perpendicular to the plane is $1.6M{a}^2$. The moment of Inertia about the tangent $\text{AB}$ in the plane of lamina is $\frac{n}{5}M{a}^2$. The value of $n$ .(only integer)

Q. Find the moment of inertia of ring having mass $2\text{kg}$ and radius $1\text{m}$ about an axis passing along the diameter of the ring.

1. $2{\text{kg m}}^2$
2. $1{\text{kg m}}^2$
3. $0.5{\text{kg m}}^2$
4. $4{\text{kg m}}^2$

Q. Define moment of inertia. State its SI unit and dimensions.

Q. $\text{MOI}$ of a circular ring of radius $R$ about the line passing through its centre of mass in the plane of circle is $I$. Find the $\text{MOI}$ of ring about a line passing through its $\text{COM}$ and perpendicular to its plane.

1. $I$
2. $\frac{I}{2}$
3. $\frac{I}{4}$
4. $2I$

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

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. 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. Find the moment of inertia of a uniform rectangular plate of mass $M$ and edges of length $l$ and $b$ about its axis passing through the centre and perpendicular to it

1. $\frac{M{l}^2}{12}$
2. $\frac{M{b}^2}{12}$
3. $\frac{M(l+b{)}^2}{12}$
4. $\frac{M(l^2+b^2)}{12}$

Q.

If the kinetic energy of a body increases by 0.1%, the percent increase of its momentum will be

1. 0.05%

2. 0.1%

3. 1.0%

4. 10%