Moment of Inertia of a Rod

Q.

Explain what is the radius of gyration?

Q.

How to derive the formula for the moment of inertia of a disc about an axis passing through its center and perpendicular to its plane?

Q. What are inertial and non-inertial frames of reference?

Q.

$1$ cusec is equal to how many litres?

Q.

A ring is hung on a nail. It can oscillate, without slipping or sliding

$\left(i\right)$ In its plane with a time period $T_1$ and,

$\left(ii\right)$ Back and forth in a direction perpendicular to its plane, with a period$T_2$.

The ratio $\frac{T_1}{T_2}$ will be:

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

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

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

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

Q.

The moment of inertia of a square lamina about the perpendicular axis through its center of mass is 20 kg per meter square then its moment of inertia about an axis touching its side and in the plane of the lamina will be

Q.

Find the center of mass of uniform semicircular ring of radius $R$ and mass $m$

Q. What is an inertial frame of reference?

Q. A thin rod of length $4l$, mass $4m$ is bent at the points shown in the figure. What is the moment of inertia of the rod about the axis passing point $\text{O}$ and perpendicular to the plane of the paper?

1. $\frac{M{l}^2}{3}$
2. $\frac{M{l}^2}{12}$
3. $\frac{10M{l}^2}{3}$
4. $\frac{M{l}^2}{24}$

Q. Three identical rods, each of mass $m$ and length $l$, form an equilateral triangle. Moment of inertia about one of the sides is

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

Q. If $I_1$ is the moment of inertia of a thin rod about an axis perpendicular to its length and passing through its centre of mass and $I_2$ is the moment of inertia of the ring (formed by bending the rod) about an axis perpendicular to the plane of ring and passing through its centre, then the ratio $\frac{I_1}{I_2}$ is

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

Q. A 6 mm fillet weld is provided as shown below. The safe load (in kN) that can be transmitted by the fillet welded joint is

[Take permissible stress for fillet weld = 108 MPa]

1. 70
2. 104
3. 128
4. 15

Q.

A ring of mass $M$ and radius $R$ is rotating with angular speed about a fixed vertical axis passing through its centre $O$ with two point masses each of mass $\frac{M}{8}$ at rest at $O$. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of system is $\frac{8}{9}\omega$ and one of the masses is at a distance of $\frac{3}{5}R$ from $O$. At this instant the distance of the other mass from $O$ is

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

Q. The moment of inertia of a uniform rod of length $2l$ and mass $m$ about an axis $(X-X)$ passing through its centre and inclined at an angle $\alpha ={30}^{\circ }$ is

1. $\frac{m{l}^2}{12}$
2. $\frac{m{l}^2}{3}$
3. $\frac{m{l}^2}{6}$
4. $\frac{m{l}^2}{4}$

Q. A rod of length $l$ having uniformly distributed charge $Q$ is rotated about one end with constant frequency $f$. Its magnetic moment is

1. $\pi fQ{l}^2$
2. $\frac{\pi fQ{l}^2}{3}$
3. $\frac{2\pi fQ{l}^2}{3}$
4. $2\pi fQ{l}^2$

Q. An angular moment of a satellite revolving round the earth in a circular orbit at a height $R$ above the surface is $L$. Here $R$ is radius of the earth.The magnitude of angular momentum of another satellite of the same mass revolving very close to the surface of the earth is

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

Q. A uniform rod of mass $M$ and length $L$ is pivoted at one end such that it can rotate in a vertical plane. There is negligible friction at the pivot. The free end of the rod is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle $\theta$ with the vertical is:

1. $g\sin \theta$
2. $\frac{g}{L}\sin \theta$
3. $\frac{3g}{2L}\sin \theta$
4. $6gL\sin \theta$

Q. Four identical rods are joined end to end to form a square. The mass of each rod is $M$. The moment of inertia of the square about the central line $y{y}^{\prime }$ as shown in figure is

1. $\frac{M{l}^2}{3}$
2. $\frac{M{l}^2}{6}$
3. none of these
4. $\frac{M{l}^2}{4}$

Q. Two thin rod of mass $m$ and length $l$ are joined together to make a ${}^{\prime }{L}^{\prime }$ shaped structure. Find out moment of inertia of structure about an axis passing through their common joining point and perpendicular to the plane of structure.

1. $\frac{m{l}^2}{12}$
2. $\frac{m{l}^2}{3}$
3. $\frac{2m{l}^2}{3}$
4. $\frac{m{l}^2}{6}$

Q.

Calculate the torque on the square plate of the previous problem if it rotates about a diagonal with the same angular acceleration.

Q. Two rods $OA$ and $OB$ of equal length and mass are lying on the $xy$ plane as shown in figure. Let $I_x,I_y$ and $I_z$ be the moment of inertia of the system about $x,y$ and $z$ axis respectively. Then,

1. $I_x=I_y>I_z$
2. $I_x>I_y>I_z$
3. $I_z>I_y>I_x$
4. $I_x=I_y<I_z$

Q. A bar magnet of moment of inertia $49\times {10}^{-2}{\text{kg m}}^2$ oscillates in a uniform magnetic field of induction $0.5\times {10}^{-4}\text{T}.$ The time period of oscillation is $8.8\text{s}.$ Find the magnetic moment of the bar magnet is (nearly)

1. $5000{\text{Am}}^2$
2. $3300{\text{Am}}^2$
3. $490{\text{Am}}^2$
4. $350{\text{Am}}^2$

Q. A thin uniform rod of mass $10\text{kg}$ and length $6\text{m}$ is rotating about an axis perpendicular to its plane, passing through the midpoint of the rod. Find the moment of inertia of the rod about this axis.

1. $120{\text{kg-m}}^2$
2. $30{\text{kg-m}}^2$
3. $60{\text{kg-m}}^2$
4. $360{\text{kg-m}}^2$

Q.

A man sitting in a train in motion is facing the engine. He tosses a coin up, the coin falls behind him. The train is moving:

1. forward with uniform speed

2. backward with uniform speed

3. forward with acceleration

4. forward with retardation

Q.

Which of the following relations is wrong?

1. Torque = Moment of inertia × Angular acceleration

2. Torque = Magnetic moment × Magnetic field

3. Moment of inertia = Torque × Angular acceleration

4. Linear momentum = Moment of inertia × Angular velocity

Q.

Water leaks out from an open tank through a hole of area 2 $m{m}^2$ in the bottom. Suppose water is filled up to a height of 80 cm and the area of cross section of the tank is 0.4 $m^2$. The pressure at the open surface and at the hole are equal to the atmospheric pressure. Neglect the small velocity of the water near the open surface in the tank. (a) Find the initial speed of water coming out of the hole. (b) Find the speed of water coming out when half of water has leaked out. (c) Find the volume of water leaked out during a time interval dt after the height remained is h. Thus find the decrease in height dh in terms of h and dt. (d) From the result of part (c) find the time required for half of the water to leak out.

Q.

A ball of radius $13$ has a round hole of radius $6$ drilled through its center. find the volume of the resulting solid.

Q. An insulating thin rod of length $l$ has a linear charge density $\rho \left(x\right)={\rho }_0\frac{x}{l}$ on it. The rod is rotated about an axis passing through the origin $(x=0)$ and perpendicular to the rod. If the rod makes $n$ rotations per second, then the time averaged magnetic moment of the rod is:

1. $\frac{\pi }{4}n\rho l^3$
2. $n\rho l^3$
3. $\frac{\pi }{3}n\rho l^3$
4. $\pi n\rho l^3$

Q. Two identical circular rods of same diameter and same length are subjected to same magnitude of axial tensile force. One of the rods is made out of mild steel having the modulus of elasticity of 206 GPa. The other rod is made out of a cast iron having the modulus of elasticity of 100 GPa. Assume both the materials to be homogenous and isotropic and the axial force causes the same amount of uniform stress in both the rods. The stresses developed are within the proportional limit of the respective materials. Which of the following observations is correct?

1. Both rods elongate by the same amount
2. Mild steel rod elongates more than the cast iron
3. Cast iron rod elongates more than the mild steel rod
4. As the stresses are equal strains are also equal in both the rods

Q. A thin uniform rod of mass $10\text{kg}$ is rotating about an axis passing through the end point of the rod of length $6\text{m}$ as shown in figure. Find the moment of inertia of the rod about this axis.

1. $43.2{\text{kg-m}}^2$
2. $24{\text{kg-m}}^2$
3. $12{\text{kg-m}}^2$
4. $38{\text{kg-m}}^2$