Parallel Axis Theorem

Q. Moment of inertia of a cylinder of mass $M,$ length $L$ and radius $R$ about an axis passing through its centre and perpendicular to the axis of the cylinder is $I=M(\frac{R^2}{4}+\frac{L^2}{12})$. If such a cylinder to be made for a given mass of a material, the ratio $\frac{L}{R}$ for it to have minimum possible $I$ is-

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

Q.

Check the correctness of the equation, $\mathrm{F}=\frac{{\mathrm{mv}}^2}{\mathrm{r}}$ where, $\mathrm{F}$ is force, $\mathrm{m}$ is mass, $\mathrm{v}$ is velocity, and $\mathrm{r}$ is radius.

Q.

On what factor does moment of inertia depends?

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

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

Q. The moment of inertia of a hollow cubical box of mass $M$ and side ${}^{\prime }{a}^{\prime }$ about an axis passing through the centers of two opposite faces is equal to

1. $\frac{5M{a}^2}{12}$
2. $\frac{5M{a}^2}{18}$
3. $\frac{5M{a}^2}{6}$
4. $\frac{5M{a}^2}{3}$

Q. Three rods each of mass $m$ and length $L$ are joined to form an equilateral triangle as shown in the figure. What is the moment of inertia about an axis passing through the centre of mass of the system and perpendicular to the plane?

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

Q. A square is made by joining four rods each of mass $m$ and length $L$ each. Its moment of inertia about an axis $PQ,$ on its plane and passing through one of its corners (as shown in the figure) will be:

1. $\frac{4}{3}{mL}^2$
2. $\frac{8}{3}{mL}^2$
3. $\frac{10}{3}{mL}^2$
4. $6{mL}^2$

Q. A lamina is made by removing a small disc of diameter $2R$ from a bigger disc of uniform mass density of radius $2R$ as shown in the figure. The moments of inertia of this lamina about an axis passing through $\text{O}$ and $\text{P}$ are $I_0$ and $I_P$ respectively. Both these axes are perpendicular to the plane of the lamina. The ratio $\frac{I_P}{I_O}$ is

1. $\frac{37}{8}$
2. $\frac{37}{13}$
3. $\frac{8}{13}$
4. $\frac{1}{4}$

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. Shown in the figure, is a hollow ice-cream cone (it is open at the top). If its mass is $M$, radius of its top is $R$ and height $H$ then, its moment of inertia about its axis is:

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

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.

Is moment of inertia a scalar quantity

Q. The MOI of a uniform circular disc is maximum about an axis perpendicular to the disc and passing through point

1. $B$
2. $A$
3. $C$
4. $O$

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

Q. Moment of inertia $I$ of a solid sphere about an axis parallel to a diameter and at a distance $x$ from its centre of mass varies as

1. 
2. 
3. 
4. 

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. 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 circular disc of mass $M$ and radius $R$is rotating about its axis with angular speed ${\omega }_1$ . If another stationary disc having radius$R/2$ and same mass $M$ is dropped co-axially on to the rotating disc. Gradually both discs attain constant angular speed ${\omega }_2$. The energy lost in the process is$p\%$ of the initial energy. Value of $p$ is __________.

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. 
2. 
3. 
4. 

Q. Four rings each of mass $M$ and radius $R$ are arranged as shown in the figure. The moment of inertia of the system about the axis yy' is

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

Q.

The moment of inertia of a uniform square plate of mass M and edge of length l about its axis passing through P and perpendicular to it is

1. 

2. 

3. 

4. 

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. Figure shows the variation of the moment of inertia of a uniform rod, about an axis passing through its centre and inclined at an angle $\theta$ to the length. The moment of inertia of the rod about an axis passing through one of its ends and making an angle $\theta =\frac{\pi }{3}$ is ( Take the figure to be a part of sine curve)

1. $0.45kg{m}^2$
2. $1.8kg{m}^2$
3. $2.4kg{m}^2$
4. $1.5kg{m}^2$

Q. A square plate of edge $a/2$ is cut out from a uniform square plate of edge $a$ as shown in the figure. The mass of the remaining portion is $M$ . The moment of inertia of the shaded portion about an axis passing through $O$ (centre of the square of side $a$ ) and perpendicular to plane of the plate is

1. $\frac{9}{64}M{a}^2$
2. $\frac{3}{16}M{a}^2$
3. $\frac{5}{12}M{a}^2$
4. $\frac{1}{6}M{a}^2$

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 circular disc of radius $b$ has a hole of radius $a$ at its centre (see figure). If the mass per unit area of the disc varies as $\left(\frac{{\sigma }_0}{r}\right)$, then the radius of gyration of the disc about its axis passing through the centre is :

1. $\sqrt{\frac{a^2+b^2+ab}{3}}$
2. $\frac{a+b}{3}$
3. $\frac{a+b}{2}$
4. $\sqrt{\frac{a^2+b^2+ab}{2}}$

Q.

A bar magnet is oscillating in the earths magnetic field for a period $T$. What happens to its period and motion if its mass is quadrupled?

1. Motion remains S.H.M. with a time period $=2T$

2. Motion remains S.H.M. with a time period $=4T$

3. Motion remains S.H.M. and the period remains nearly constant

4. Motion remains S.H.M. with a time period $=\frac{T}{2}$

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. 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. Three identical circular planner disc each of mass $m$ and radius $r$ are welded as shown in figure. The moment of inertia of the arrangement about the axis normal to the plane and passing through the point $P$ is

1. $9m{r}^2$
2. $\frac{1}{2}m{r}^2$
3. $\frac{9}{2}m{r}^2$
4. $\frac{19}{2}m{r}^2$