Parallel Axis Theorem

Q.

Consider a uniform wire of mass $M$ and length $L$. It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is

1. $\frac{1}{2}\frac{M{L}^2}{{\mathrm{\pi }}^2}$

2. $\frac{2}{5}\frac{M{L}^2}{{\mathrm{\pi }}^2}$

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

4. $\frac{M{L}^2}{{\mathrm{\pi }}^2}$

Q. Four thin rods of same mass 'M' and same length 'l', form a square as shown in the figure. Moment of inertia of this system about an axis through centre O and perpendicular to its plane is

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

Q.

Calculate the moment of inertia of a thin uniform rod of the mass $100\mathrm{g}$and length$60\mathrm{cm}$ about an axis perpendicular to its length and passing through

(a) its center

(b) one end.

Q. Find the MOI of four identical solid spheres about axis $X{X}^{\prime }$ placed in horizontal plane each of mass $M$ and radius $R$ as shown in figure. Line $X{X}^{\prime }$ touches two spheres (3 and 4) and passes through diameter of other two spheres (1 and 2).

1. $\frac{9}{10}M{R}^2$
2. $\frac{5}{18}M{R}^2$
3. $\frac{18}{5}M{R}^2$
4. $\frac{9}{5}M{R}^2$

Q. A square lamina of mass $m=4\text{kg}$ having sides of length $a=2\text{m}$. Find the moment of inertia of the lamina along one of its side.

1. $\frac{4}{3}{\text{kg-m}}^2$
2. $\frac{8}{3}{\text{kg-m}}^2$
3. $\frac{16}{3}{\text{kg-m}}^2$
4. $\frac{2}{3}{\text{kg-m}}^2$

Q. A square plate of edge ${\frac{{}^{\prime }a}{2}}^{\prime }$ is cut from a uniform square plate of edge 'a' as shown in figure. Mass of square plate of edge 'a' is $M$. Find out the moment of inertia of the remaining square plate about an axis passing through 'O' (center of square plate of side 'a') and perpendicular to the plane of the plate.

1. $\frac{5}{192}M{a}^2$
2. $\frac{27}{64}M{a}^2$
3. $\frac{9}{64}M{a}^2$
4. $\frac{M{a}^2}{6}$

Q. Find the MOI of four identical solid spheres each of mass $M$ and radius $R$ placed in a horizontal plane as shown in figure. Line $X{X}^{\prime }$ touches two spheres and passes through the diameter of the other two spheres.

1. $\frac{9}{10}M{R}^2$
2. $\frac{5}{18}M{R}^2$
3. $\frac{18}{5}M{R}^2$
4. $\frac{9}{5}M{R}^2$

Q. The moment of inertia of a hollow cylinder of radius $R$ and mass $M$ about an axis passing through the outer circumference along the height of the hollow cylinder is

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

Q. A thin square plate of side $3\text{m}$ has mass $5\text{kg}$. Find the moment of inertia about axis $AB$ as shown in figure.

1. $15{\text{kg-m}}^2$
2. $30{\text{kg-m}}^2$
3. $45{\text{kg-m}}^2$
4. $7.5{\text{kg-m}}^2$

Q. The moment of inertia of a rod of length 'l' about an axis passing through its centre of mass and perpendicular to rod is 'I'. The moment of inertia of hexagonal shape formed by six such rods, about an axis passing through its centre of mass and perpendicular to its plane will be

1. $16l$
2. $40l$
3. $60l$
4. $80l$

Q. Three rings each of mass $M$ and radius $R$ are arranged as shown. The moment of inertia of the system about $YY\text{'}$ 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. A rectangular lamina $ABCD$ of mass $m=3\text{kg}$ and sides $AB=2\text{m}$ and $BC=4\text{m}$ is shown in the figure. Find its moment of inertia about an axis passing through point $A$ and perpendicular to the plane of lamina.

1. $5{\text{kg.m}}^2$
2. $20{\text{kg.m}}^2$
3. $15{\text{kg.m}}^2$
4. $10{\text{kg.m}}^2$

Q. The moment of inertia of a hollow cylinder of radius $R$ and mass $M$ about an axis passing through the outer circumference along the height of the hollow cylinder is

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

Q. Four identical thin rods of mass $m$ and length $l$ each, form a square frame as shown in figure. Find the MOI of the frame about an axis passing through one of its corners and perpendicular to its plane.

1. $\frac{10m{l}^2}{3}$
2. $\frac{4m{l}^2}{3}$
3. $\frac{2m{l}^2}{5}$
4. $\frac{m{l}^2}{3}$

Q. Necessary condition for the application of parallel axes theorem:
$I_A=I_B+M{d}^2$
where $A\text{and}B$ are the axes of rotation of body and $M$ is the mass of body.

1. Axis $B$ must pass through C.O.M of body
2. Axis $A$ must be parallel to axis $B$
3. $d$ must be shortest distance between axis $A$ and axis $B$
4. All of these

Q. The moment of inertia of a rod of length $l$ about an axis passing through its centre of mass and perpendicular to the rod is $I$. The moment of inertia of hexagonal shape formed by six such rods, about an axis passing through its centre of mass and perpendicular to its plane will be

1. $16I$
2. $40I$
3. $60I$
4. $80I$

Q. A uniform cylinder has a radius $R$ and length $L$. If the moment of inertia of this cylinder about an axis passing through its centre and normal to its circular face is equal to the moment of inertia of the same cylinder about an axis passing through its centre and normal to its length; then

1. $L=R$
2. $L=\sqrt{3}R$
3. $L=\frac{R}{\sqrt{3}}$
4. $L=0$

Q. A system of solid discs, each of mass $M$ and radius $R$ is shown in figure.


Find the MOI about an axis passing through point $\text{O}$ and perpendicular to the plane.

1. $\frac{9M{R}^2}{2}$
2. $9M{R}^2$
3. $18M{R}^2$
4. $27M{R}^2$

Q. The moment of inertia of a solid sphere, about an axis parallel to its diameter and at a distance of $x$ from it is $I\left(x\right)$. Which one of the graphs represents the variation of $I\left(x\right)$ with $x$ correctly ?

1. 
2. 
3. 
4. 

Q. The moment of inertia of a hollow cylinder of radius $R$ and mass $M$ about an axis passing through the outer circumference along the height of the hollow cylinder is

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

Q. Two thin discs each of mass 4.0 kg and radius 0.4 m are attached as shown in figure to form a rigid body. The rotational inertia of this body about an axis perpendicular to the plane of disc $B$ and passing through its center is.

1. $1.6kg{m}^2$
2. $3.2kg{m}^2$
3. $6.4kg{m}^2$
4. $0.8kg{m}^2$

Q. Two thin discs each of mass 4.0 kg and radius 0.4 m are attached as shown in figure to form a rigid body. The rotational inertia of this body about an axis perpendicular to the plane of disc $B$ and passing through its center is.

1. $1.6kg{m}^2$
2. $3.2kg{m}^2$
3. $6.4kg{m}^2$
4. $0.8kg{m}^2$

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

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

Q. Find the M.O.I $\left(I\right)$ of a uniform solid sphere of radius $R$ and mass $M$ about its tangent

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

Q. M.O.I. of a ring of radius $R$ and mass $M$ about a line passing throught its center and perpendicular to its plane is $I$. Find out the M.O.I $\left(I_t\right)$ about an axis which is a tangent and also perpendicular to the plane of circle as shown in figure.

1. $I+M{R}^2$
2. $I-M{R}^2$
3. $I-MR$
4. $I+MR$

Q. There are four solid balls with their centres at the four corners of a square of side $a$. The mass of each sphere is $m$ and radius $r$. Find the moment of inertia of the system about one of the sides of the square.

1. $\frac{8}{5}m{r}^2+2m{a}^2$
2. $\frac{5}{8}m{r}^2+2m{a}^2$
3. $m{r}^2+\frac{8}{5}m{a}^2$
4. $m{r}^2+\frac{5}{8}m{a}^2$

Q. A disc of mass $M$ and radius $R$ is attached to a rectangular plate of the same mass $M$, breadth $R$ and length $2R$ as shown in figure. The moment of inertia of the system about the axis $AB$ passing through the centre of the disc and on the plane is $I=\frac{1}{\alpha }\left(\frac{31}{3}M{R}^2\right)$. Then, the value of $\alpha$ is

Q. For the thin rod of mass $m$ shown in the figure, choose the correct statement(s) regarding moment of inertia about the axis $1,2,3,4$. Here axis $4$ is passing through the centre of the rod.

1. $\text{M.O.I}$ about axis $1$ is zero.
2. $\text{M.O.I}$ about axis $2$ is $\frac{m{L}^2}{3}$
3. $\text{M.O.I}$ about axis $3$ is $\frac{13m{L}^2}{12}$
4. $\text{M.O.I}$ axis $4$ is $\frac{m{L}^2}{12}$

Q. Four holes of radius $R$ each are cut from a thin square plate of side $4R$ and mass $M$. The moment of inertia of the remaining portion about $z-$ axis (out of the plane) is

1. $\frac{\pi }{12}M{R}^2$
2. $(\frac{4}{3}-\frac{\pi }{4})M{R}^2$
3. $(\frac{8}{3}-\frac{10\pi }{16})M{R}^2$
4. $(\frac{4}{3}-\frac{\pi }{6})M{R}^2$

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