Rotational Inertia

Q.

In a rectangle ABCD, BC = 2 AB. The moment of inertia is minimum along axis through

1. CD

2. BC

3. HF

4. EG

Q.

Four particles each of mass $m$ are kept at the four corners of a square of edge $a$. Find the moment of inertia of the system about the line perpendicular to the plane of the square and passing through the centre of the square

1. 

2. 

3. 

4. 

Q. Four particles each of mass m are arranged at the corners of a massless square of side a. The MI of system about an axis which is perpendicularto the plane of square and passes through one corner is

Q.

The radius of a solid sphere is R and its density D. When it is made to rotate about an axis passing through any diameter of sphere, expression for its moment of inertia is

1. 

2. 

3. 

4. 

Q. Due to global warming, ice on polar caps is likely to melt in larger quantity. Due to this effect

1. Moment of inertia of earth shall decrease
2. Angular momentum of earth shall decrease
3. Angular velocity of earth shall increase
4. Length of the day shall increase

Q. 56.What is the moment of inertia of a plane sheet??like rectangular or a plane square sheet

Q. A CLOSED CYLINDRICAL TUBE CONTAINS SOME WATER LIES IN AHORIZONTAL PLANE IF TUBE IS ROTATED ABOUT A PERPENDICULAR BISECTOR THE MOMENT OF INERTIA OF WATER ABOUT THE AXIS\lbrack

Q.

Given the magnitude of vector A to be 6 units. Find the component of A along the x-axis?

1. 

2. 

3. 

4. 

Q. state true or false (and explain for false statement) 1.a rigid body must be in equilibrium under action of 2 equal and opposite forces 2.as star collapses its angular velocity increases because the mass of star increases 3. a rotating body can be in stable and instable equilibrium because moment of inertia is always different for different axis of rotation 4.for the purpose of calculation of moment of inertia , bodys mass can be assumed to be concentrated at its centre of mass a. 1 and 2 b. 2 and 3 c.all d.none

Q. Two balls each of mass $1\text{kg}$ are connected to both ends of a uniform circular rod of mass $5\text{kg}$ and length $2\text{m}$. Find the moment of inertia of the system about an axis passing through the middle of the rod and perpendicular to the length of the rod.

1. $\frac{5}{3}{\text{kg-m}}^2$
2. $2{\text{kg-m}}^2$
3. $\frac{13}{3}{\text{kg-m}}^2$
4. $\frac{11}{3}{\text{kg-m}}^2$

Q. The moment of inertia of a body does not depend on

1. None of the above
2. Mass of the body
3. The distribution of the mass in the body
4. Position of axis of rotation of the body.

Q. A square plate of edge $\frac{a}{2}$ is cut out from a uniform square plate of edge ${}^{\prime }{a}^{\prime }$ as shown in the figure. The mass of the remaining portion is $M$. The moment of inertia of the remaining portion about an axis passing through $\text{'O'}$ (centre of the square of side $a$) and perpendicular to the 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. $\frac{9}{16}M{a}^2$

Q. Point masses of $1,2,3$ and $4\text{kg}$ are lying at the points $(0,0,0),(2,0,0),(0,3,0)\text{and}(-2,-2,0)$ respectively. What will be the moment of inertia of the system about $x$ - axis?

1. $51{\text{kg m}}^2$
2. $43{\text{kg m}}^2$
3. $67{\text{kg m}}^2$
4. $40{\text{kg m}}^2$

Q. The moment of inertia of uniform cylinder of length L and radius R about is perpendicular bisector is I.What is the ratio of L/R such that the moment of inertia is minimum?

Q. The radius of gyration of a uniform circular disc of radius a, rotating about an axis perpendicular to the plane and passing through its centre of mass, is

घूर्णन करती हुई a त्रिज्या की समरूप वृत्तीय डिस्क के तल के लम्बवत् तथा इसके द्रव्यमान केन्द्र से गुजरने वाले अक्ष के सापेक्ष इसकी परिभ्रमण त्रिज्या है

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

Q. Column I of the following table gives a list of bodies and column II gives their moments of inertia about the axis of symmetry passing through their centre of mass. Assume mass of each of the bodies is $M$. Then identify the correct matching.

Column-I	Column-II
i. Thin uniform hollow cylinder of radius $R$	p. $\frac{2}{5}M{R}^2$
ii. Uniform solid sphere of radius $R$	q. $\frac{2}{3}M{R}^2$
iii. Uniform rectangular lamina of length $l$ and breadth $b$	r. $M{R}^2$
iv. Thin uniform hollow sphere of radius $R$	s. $M\left(\frac{l^2+b^2}{12}\right)$

1. $i-r,ii-q,iii-s,iv-p$
2. $i-r,ii-p,iii-s,iv-q$
3. $i-r,ii-p,iii-s,iv-p$
4. $i-r,ii-q,iii-s,iv-q$

Q. Find the moment of inertia of a non-uniform rod having mass per unit length $\lambda ={\lambda }_0(1+\frac{x}{L})$ about an axis $\perp$ to the length of the rod and passing through an end point $L$. Length of the rod is $L$ and $m$ is mass of the rod.

1. $\frac{m{L}^2}{3}$
2. $\frac{7m{L}^2}{12}$
3. $\frac{7m{L}^2}{18}$
4. $\frac{m{L}^2}{18}$

Q. A uniform rectangular lamina is shown in figure. Find the moment of inertia $I_{\text{com}}$ about $z$ axis which passes through its centre and perpendicular to its plane. Here $b=10\text{m}$, $l=20\text{m}\&M=12\text{kg}$.

1. $100{\text{kg-m}}^2$
2. $1200{\text{kg-m}}^2$
3. $400{\text{kg-m}}^2$
4. $500{\text{kg-m}}^2$

Q. Find the moment of inertia of a non-uniform rod having mass per unit length $\lambda ={\lambda }_0(1+\frac{x}{L})$ about an axis $\perp$ to the length of the rod and passing through an end point $L$. Length of the rod is $L$ and $m$ is mass of the rod.

1. $\frac{m{L}^2}{3}$
2. $\frac{7m{L}^2}{12}$
3. $\frac{7m{L}^2}{18}$
4. $\frac{m{L}^2}{18}$

Q.

Two particles having masses $m_1$ and $m_2$are situated in a plane perpendicular to line AB at a distance of$r_1$and $r_2$ respectively as shown.

Find the moment of intertia about axis passing through centre of mass and perpendicular to line joining $m_1$ and $m_2$

1. $\frac{m_1{m}_2}{m_1+m_2}$ $(r_1+r_2{)}^2$

2. $\frac{m_1{m}_2}{m_1+m_2}$ $(r_1-r_2{)}^2$

3. $\frac{m_1{m}_2}{m_1+m_2}$ $(r_1^2+r_2^2)$

4. $\frac{m_1{m}_2}{m_1+m_2}$ $(r_1^2+r_2^2)$

Q. Point masses of $1,2,3$ and $4\text{kg}$ are lying at the points $(0,0,0),(2,0,0),(0,3,0)\text{and}(-2,-2,0)$ respectively. What will be the moment of inertia of the system about $x$ - axis?

1. $51{\text{kg m}}^2$
2. $43{\text{kg m}}^2$
3. $67{\text{kg m}}^2$
4. $40{\text{kg m}}^2$

Q. A uniform thin rod of length $L$ and mass $M$ is bent at the middle point $O$ as shown in figure. Consider an axis passing through its middle point $O$ and perpendicular to the plane of the bent rod. Then moment of inertia about this axis is :

1. $\frac{2}{3}m{L}^2$
2. $\frac{1}{3}m{L}^2$
3. $\frac{1}{12}m{L}^2$
4. dependent on $\theta$

Q. Three particles each of mass $1\text{kg}$ are placed at the vertices of an equilateral triangle $ABC$ of side $2\text{m}$. Find the moment of inertia of system about a line perpendicular to the plane of system and passing through middle point ($D$) of side $BC$ of the triangle.

1. $5{\text{kg-m}}^2$
2. $3{\text{kg-m}}^2$
3. $2.5{\text{kg-m}}^2$
4. $1.1{\text{kg-m}}^2$

Q. A paraboloid shaped solid object is formed by rotation on parabola y=2x^2 about y axis as shown . If height of body is h then position of center of mass from origin is

Q. A wheel of the moment of inertia $10kg-m^2$ is rotating at $10$ rotations per minute. Find the work done in increasing its speed to $5$ times of its initial value.

Q. Three particles of masses $300g,800g$ and $700g$ are placed at positions $A(2,4),B(3,5)$ and $C(-3,-1)$ respectively. What is the moment of inertia of the system about the origin?

1. $40.2{\text{kg-m}}^2$
2. $42{\text{kg-m}}^2$
3. $34{\text{kg-m}}^2$
4. $30{\text{kg-m}}^2$

Q. Suppose there existed a planet that went around the sun twice as fast as the earth.What would be its orbital size as compared to that of the earth ?

Q. Find the moment of inertia of a pair of spheres, each having a mass mass m and radius r, kept in contact about the tangent passing through the point of contact.

Q. if moment of inertia of a disc is I. if we rake only 1/4th of the disc keeping axis same, then its moment of inerrtia will have what valu

Q. Density of a solid sphere is given by the expression $\rho ={\rho }_0(1+\frac{x}{R})$ where $x$ is the distance from the centre of the sphere and $0\le x\le R$, $R$ is the radius of the solid sphere. Moment of inertia of the solid sphere about an axis passing through its centre will be: (Given ${\rho }_0$ is a constant)

1. $\frac{44}{45}\pi {\rho }_0{R}^5$
2. $\frac{44}{45}\pi {\rho }_0{R}^2$
3. $\frac{44}{35}\pi {\rho }_0{R}^2$
4. $\frac{48}{38}\pi {\rho }_0{R}^5$