Moment of Inertia of a Disc

Q. Find the moment of inertia of an equilateral triangular lamina of mass $1\text{kg}$ and having sides $3\text{m}$ about the axis passing through its centre of mass and perpendicular to the plane of triangular lamina.

1. $0.75{\text{kg-m}}^2$
2. $1.5{\text{kg-m}}^2$
3. $1{\text{kg-m}}^2$
4. $3{\text{kg-m}}^2$

Q. Two discs are rotating about the same axis passing through their centres and perpendicular to both disc's planes. If disc $1$ has mass $5\text{kg}$ and $30\text{cm}$ radius and disc $2$ has mass $7\text{kg}$ and $60\text{cm}$ radius, then find the moment of intertia about an axis $\text{OO'}$.

1. $1.26{\text{kg-m}}^2$
2. $1.96{\text{kg-m}}^2$
3. $1.48{\text{kg-m}}^2$
4. $1.16{\text{kg-m}}^2$

Q. The surface mass density of a disc of radius $a$ varies with radial distance $r$ as $\sigma =A+Br$ where A & B are positive constants. Then, moment of inertia of the disc about an axis passing through its centre and perpendicular to the plane (in ${\text{kg m}}^2$) is:

1. $2\pi a^4(\frac{A}{4}+\frac{Ba}{5})$
2. $2\pi a^4(\frac{Aa}{4}+\frac{B}{5})$
3. $\pi a^4(\frac{A}{4}+\frac{Ba}{5})$
4. $2\pi a^4(\frac{A}{5}+\frac{Ba}{4})$

Q. Two circular discs A and B are of equal masses and thickness but made of metals with densities $d_A$ and $d_B(d_A>d_B).$ If their moments of inertia about an axis passing through centres and normal to the circular faces be $I_A$ and $I_B$, then

1. $I_A=I_B$
2. $I_A>I_B$
3. $I_A<I_B$
4. $I_A>=<I_B$

Q. The surface mass density of a disc of radius $a$ varies with radial distance $r$ as $\sigma =A+Br$ where A & B are positive constants. Then, moment of inertia of the disc about an axis passing through its centre and perpendicular to the plane (in ${\text{kg m}}^2$) is:

1. $2\pi a^4(\frac{A}{4}+\frac{Ba}{5})$
2. $2\pi a^4(\frac{Aa}{4}+\frac{B}{5})$
3. $\pi a^4(\frac{A}{4}+\frac{Ba}{5})$
4. $2\pi a^4(\frac{A}{5}+\frac{Ba}{4})$

Q. A thin disc of mass M and radius R has mass per unit area $\sigma \left(r\right)=k{r}^2$, where r is the distance from its centre. Its moment of inertia about an axis going through its centre of mass and perpendicular to its plane is

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

Q. Find the moment of inertia of an equilateral triangular lamina of mass $1\text{kg}$ and having sides $3\text{m}$ about the axis passing through its centre of mass and perpendicular to the plane of triangular lamina.

1. $0.75{\text{kg-m}}^2$
2. $1.5{\text{kg-m}}^2$
3. $1{\text{kg-m}}^2$
4. $3{\text{kg-m}}^2$

Q. A cylinder of mass $M$ has a length $L$ that is $\sqrt{3}$ times its radius $R.$ What is the ratio of its moment of inertia about its own axis and that about an axis passing through its centre and perpendicular to its axis?

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

Q. Two discs have same mass and thickness. Their materials are of densities $d_1$ and $d_2$. The ratio of their moments of inertia about an axis passing through the centre and perpnedicular to the plane is

1. $d_1:d_2$
2. $d_2:d_1$
3. $\left(\frac{d_1}{d_2}\right)$
4. ${\left(\frac{d_2}{d_1}\right)}^2$

Q. Two discs are rotating about the same axis passing through their centres and perpendicular to both disc's planes. If disc $1$ has mass $5\text{kg}$ and $30\text{cm}$ radius and disc $2$ has mass $7\text{kg}$ and $60\text{cm}$ radius, then find the moment of intertia about an axis $\text{OO'}$.

1. $1.26{\text{kg-m}}^2$
2. $1.96{\text{kg-m}}^2$
3. $1.48{\text{kg-m}}^2$
4. $1.16{\text{kg-m}}^2$

Q. Two circular discs $A$ and $B$ are of equal masses and thickness but made of metals with densities $d_A$ and $d_B(d_A>d_B)$ . If their moments of inertia about an axis passing through their centres and perpendicular to the circular faces are $I_A$ and $I_B$ then

1. $I_A=I_B$
2. $I_A>I_B$
3. $I_A<I_B$
4. $I_A\ge I_B$

Q. The uniform disc shown in the figure has a moment of inertia of $0.6{\text{kg-m}}^2$ around the axis that passes through $O$ and is perpendicular to the plane. If a segment is cut out from the disc as shown, what is the moment of inertia of the remaining disc in ${\text{kg-m}}^2$ ?

Q. Find the moment of inertia of an equilateral triangular lamina of mass $1\text{kg}$ and having sides $3\text{m}$ about the axis passing through its centre of mass and perpendicular to the plane of triangular lamina.

1. $0.75{\text{kg-m}}^2$
2. $1.5{\text{kg-m}}^2$
3. $1{\text{kg-m}}^2$
4. $3{\text{kg-m}}^2$

Q. A circular disc X of radius 'R' is made from an iron plate of thickness 't', and another disc Y of radius '4R' is made from an iron plate of thickness '$\frac{t}{4}$'. Then the relation between the moment of inertia $I_x$ and $I_y$ is

1. $I_y=64I_x$
2. $I_y=32I_x$
3. $I_y=16I_x$
4. $I_y=I_x$

Q. A thin uniform disc of mass $M$ and radius $R$ has concentric hole of radius $r$. Find the moment of inertia of the disc about an axis passing through its centre and perpendicular to its plane.

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

Q. A circular disc of radius $1\text{m}$ is rotating about an axis passing through its $COM$ and perpendicular to its plane. The moment of inertia about this axis is given by $I_{com}={\text{2 kg-m}}^2$. If the disc is rotated about an axis along its diameter, then the moment of inertia along that axis is $I_{dia}$. The value of $(I_{com}-I_{dia})$ is equal to

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

Q. Five particles of mass 2 kg are attached to the rim of a circular disc of radius 0.1 m and negligible mass. Moment of inertia of the system about the axis passing through the center of the disc and perpendicular to its plane is.

1. $1kg{m}^2$
2. $0.1kg{m}^2$
3. $2kg{m}^2$
4. $0.2kg{m}^2$

Q. Two discs have same mass and thickness. Their materials are of densities $d_1$ and $d_2$. The ratio of their moments of inertia about an axis passing through the centre and perpnedicular to the plane is

1. $d_1:d_2$
2. $d_2:d_1$
3. $\left(\frac{d_1}{d_2}\right)$
4. ${\left(\frac{d_2}{d_1}\right)}^2$

Q. Two discs of mass $2\text{kg}$ and $3\text{kg}$ of radius $1\text{m}$ and $2\text{m}$ respectively placed together are rotated about an axis through their common centre as shown in figure. Find the net moment of inertia of the system about given axis.

1. $6{\text{kg-m}}^2$
2. $5{\text{kg-m}}^2$
3. $1{\text{kg-m}}^2$
4. $7{\text{kg-m}}^2$

Q. Two discs of mass $2\text{kg}$ and $3\text{kg}$ of radius $1\text{m}$ and $2\text{m}$ respectively placed together are rotated about an axis through their common centre as shown in figure. Find the net moment of inertia of the system about given axis.

1. $6{\text{kg-m}}^2$
2. $5{\text{kg-m}}^2$
3. $1{\text{kg-m}}^2$
4. $7{\text{kg-m}}^2$

Q. From a circular disc of radius R and mass 9M, a small disc of radius R/3 is removed from the disc. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through O is

1. $4M{R}^2$
2. $\frac{40}{4}M{R}^2$
3. $10M{R}^2$
4. $\frac{37}{9}M{R}^2$