Radius of Gyration

Q. The ratio of the radii of gyration of a circular disc to that of a circular ring, each of same mass and radius, around their respective axes is

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

Q. The ratio of radius of gyration of a solid sphere and hollow sphere of the same mass and radius about a tangential axis is

1. $5:\sqrt{21}$
2. $\sqrt{7}:\sqrt{3}$
3. $\sqrt{5}:\sqrt{6}$
4. $\sqrt{21}:5$

Q. Radius of gyration does not depend on

1. Shape and size of the body.
2. Position and configuration of axis of rotation.
3. Distribution of mass of the body with respect to axis of rotation.
4. Mass of the body.

Q. The radius of gyration of a spherical shell of mass $M$ and radius $R$ about its tangential axis will be

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

Q. The radius of gyration of a uniform rod of length $L$ about an axis passing through its centre of mass is

1. $\frac{L}{2\sqrt{3}}$
2. $\frac{L^2}{12}$
3. $\frac{L}{\sqrt{2}}$
4. $\frac{L}{\sqrt{3}}$

Q. $\left(a\right)$ Centre of gravity $(C.G.)$ of a body is the point at which the weight of the body acts.
$\left(b\right)$ Centre of mass coincides with the centre of gravity if the earth is assumed to have infinitely large radius.
$\left(c\right)$ To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its $(C.G.)$
$\left(d\right)$ The radius of gyration of any body rotating about an axis is the length of the perpendicular distance from the $(C.G.)$ of the body to the axis.

Which one of the following pairs of statements is correct?

1. (b) and (c)
2. (c) and (d)
3. (a) and (c)
4. (a) and (b)

Q. Four particles each of mass $m$ are placed at the corners of a square of side length $l$. The radius of gyration of the system about an axis perpendicular to the square and passing through its centre is

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

Q. Find the ratio of radii of gyration of a circular disc and a circular ring of same mass and radius, about an axis passing through their centre and perpendicular to their planes.

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

Q. Moment of inertia of solid hemisphere of mass M and radius R about an axis passing through centre of mass as shown, is

1. $\frac{83}{320}M{R}^2$
2. $\frac{85}{375}M{R}^2$
3. $\frac{73}{360}M{R}^2$
4. $\frac{83}{260}M{R}^2$

Q.

The radius of gyration of a uniform disc about a line perpendicular to the disc equals its radius. Find the distance of the line from the centre.

1. 

2. 

3. 

4. 

Q.

Find the radius of gyration of a circular ring of radius $r$ about a line perpendicular to the plane of the ring and passing through one of its particles.

1. 

2. 

3. R

4. 2R

Q. The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is

1. $1:\sqrt{2}$
2. $1:3$
3. $\sqrt{5}:\sqrt{6}$
4. $2:1$

Q. Find the radius of gyration of a uniform disc of radius $R$ and mass $M$ about its edge and perpendicular to the disc.

1. $\sqrt{\frac{3}{2}}R$
2. $\frac{9}{2}R$
3. $\frac{\sqrt{5}}{2}R$
4. $\frac{3}{2}R$

Q.

The moment of inertia of a solid sphere about an axis passing through the center of gravity is, $\frac{2}{5}M{R}^2$ then its radius of gyration about a parallel axis at a distance $2R$ from the first axis is

1. $5R$

2. $\sqrt{\frac{22}{5}}R$

3. $\frac{5}{2}R$

4. $\sqrt{\frac{12}{5}}R$

Q.

Gravitational force on the surface of the moon is only $\frac{1}{6}$ as strong as gravitational force on the earth. What is the weight in the newton’s of a $10kg$ object on the moon and on the earth?

Q. Radius of gyration of a body about an axis$\left(I_A\right)$ is $5\text{m}$. Perpendicular distance of $\left(I_A\right)$ from center of mass of body is $3\text{m}$. Find its radius of gyration about an axis$\left(I_B\right)$ which is parallel to $\left(I_A\right)$ and also passing through center of mass of body.

1. $4\text{m}$
2. $5\text{m}$
3. $3\text{m}$
4. $6\text{m}$

Q.

What is center of gyration?

Q.

Calculate the ratio of the angular momentum of the earth about its axis due to its spinning motion to that about the sun due to its orbital motion. Radius of the earth = 6400 km and radius of the orbit of the earth about the sun = 1.5 $\times {10}^8$ km.

Q. Given below are two statements : one is labelled as Assertion $A$ and the other is labelled as Reason $R$.

$\text{Assertion A}$ : Moment of inertia of a circular disc of mass ${}^{\prime }{M}^{\prime }$ and radius ${}^{\prime }{R}^{\prime }$ about $X,Y$ axes (passing through its plane) and $Z$-axis which is perpendicular to its plane were found to be $I_x$, $I_y$ and $I_z$ respectively. The respective radii of gyration about all the three axes will be the same.

$\text{Reason R}$ : A rigid body making rotational motion has fixed mass and shape.

In the light of the above statements, choose the most appropriate answer, from the options given below.

1. Both $A$ and $R$ are correct, and $R$ is the correct explanation of $A$.
2. Both $A$ and $R$ are correct but $R$ is NOT the correct explanation of $A$.
3. $A$ is correct but $R$ is not correct.
4. $A$ is not correct but $R$ is correct.

Q. Find the radius of gyration for a rectangular lamina having mass $m$ about an axis passing through C.O.M and perpendicular to the plane of lamina. (Given: $\text{length,}a=4\text{m};\text{breadth,}b=3\text{m})$

1. $5\sqrt{3}$
2. $\frac{5\sqrt{3}}{6}$
3. $5$
4. $\frac{5}{6}$

Q. Disc of radius $\frac{a}{2}$ is cut out from a disc of radius $a$. Find $x$ co-ordinate of centre of mass from origin.

1. $\frac{5a}{6}$
2. $\frac{a}{6}$
3. $\frac{a}{3}$
4. $\frac{a}{2}$

Q.

Draw a diagram to find the C.G of a triangular lamina.

Q. Find the ratio of radius of gyration about natural axis of a circular disk to that of a circular ring each having same mass and same radius.

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

Q. Find the radius of gyration of a uniform circular disc of radius $R=1\text{m}$ and mass $M=2\text{kg}$ about its axis passing through the edge and normal to the disc as shown in figure.

1. $1\text{m}$
2. $\sqrt{\frac{3}{2}}\text{m}$
3. $\sqrt{\frac{5}{2}}\text{m}$
4. $\sqrt{\frac{1}{2}}\text{m}$

Q.

Match the following question:

Column 1	Column 2
Acceleration due to gravity($g$)	vector quantity
Gravitational constant($G$)	$F=\frac{GMm}{r^2}$
Force of gravitation	Scalar quantity
Weight	$9.8m/s^2$

Q. If I is the moment of inertia of a ring having $\alpha$ coefficient of linear expansion, then the change in $I$ corresponding to a small change in temperature $\Delta T$ is

1. $\alpha I\Delta T$
2. $\frac{1}{2}\alpha I\Delta T$
3. $2\alpha I\Delta T$
4. $3\alpha I\Delta T$

Q. Find the radius of gyration of a thin uniform rod of mass ${}^{\prime }{m}^{\prime }$ and length $l=9\text{m}$ about an axis passing through one end and perpendicular to the rod as shown in figure.

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

Q.

A tunnel through a mountain for a four-lane highway is to have an elliptical opening.

The total width of the highway (not the opening) is to be $16\mathrm{m}$, and the height at the edge of the road must be sufficient for a truck $4\mathrm{m}$ high to clear if the highest point of the opening is to be $5\mathrm{m}$ approximately.

How wide must the opening be,

Q. A circular disc is rotating about its own axis. An external opposing torque of magnitude $0.04\text{Nm}$ is applied on the disc due to which it comes to rest in $5\text{seconds}$. The magnitude of initial angular momentum of the disc is

1. $1{\text{kg-m}}^2/\text{sec}$
2. $0.2{\text{kg-m}}^2/\text{sec}$
3. $0.4{\text{kg-m}}^2/\text{sec}$
4. $0.8{\text{kg-m}}^2/\text{sec}$

Q. Two identical solid sphere each of mass $m=5\text{kg}$ and radius $r=1\text{m}$ are joined at the ends of a light rod of length $l=2\text{m}$ to form a system as shown in figure. Radius of gyration of the system about an axis perpendicular to the length of rod and passing through center of mass of system is

1. $\sqrt{2.4}\text{m}$
2. $\sqrt{3.2}\text{m}$
3. $\sqrt{44}\text{m}$
4. $\sqrt{4.4}\text{m}$