Rotational Work and Energy

Q. Four massless springs whose force constants are 2k, 2k, k and 2k respectively are attached to mass M kept on a frictionless plane (as shown in figure). If the mass M is displaced in the horizontal direction, then the frequency of the system is

1. 
2. 
3. 
4. 

Q. A particle of mass m is projected with velocity $v$ making an angle of ${45}^{\circ }$ with the horizontal. If maximum height of the projectile is $h$, then

1. angular momentum of projectile about origin remains conserved
2. angular momentum of the projectile about origin when it just starts its motion is zero
3. angular momentum of projectile about origin when it reaches the highest point is $m\sqrt{g{h}^3}$
4. angular momentum of projectile about origin when it reaches the highest point is $m\sqrt{2g{h}^3}$.

Q. A rigid body is made up of three identical thin rods $A,B$ and $C$, each of length $L$ fastened together in the form of letter $H$. The body is free to rotate about a horizontal axis that runs along the length of one of the legs ($A$) of the body $H$. The body is allowed to fall from rest from a position in which the plane of $H$ is horizontal. What is the angular speed of the body when the plane of $H$ becomes vertical?

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

Q. A ball rolls without slipping. The radius of gyration of the ball about an axis passing through its center of mass is $k$. If radius of the ball is $R$, then the fraction of total energy associated with its rotational energy will be

1. $\frac{k^2}{k^2+R^2}$
2. $\frac{R^2}{k^2+R^2}$
3. $\frac{k^2+R^2}{R^2}$
4. $\frac{k^2}{R^2}$

Q. A magnet of magnetic moment $M$ is situated with its axis along the direction of a magnetic field of strength $B$. The work done in rotating it by an angle of ${180}^{\circ }$ will be

1. $-MB$
2. $+MB$
3. $0$
4. $+2MB$

Q.

What is the radius of gyration? On what factors its depends?

Q. A solid sphere of mass $1\text{kg}$ and radius $10\text{cm}$ rolls down an inclined plane of height $7\text{m}$. The velocity of its center as it reaches the ground level is

1. $7\text{m/s}$
2. $10\text{m/s}$
3. $15\text{m/s}$
4. $20\text{m/s}$

Q. A small block is connected to one end of a massless spring of un-stretched length \(4.9~ m\). The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by \(0.2~ m\) and released from rest at \(t = 0\). It then executes simple harmonic motion with angular frequency \(\omega = \dfrac{\pi}{3}\text{rad/s}\). Simultaneously at \(t = 0\), a small pebble is projected with speed \(v\) from point \(P\) at an angle of \(45^o\) as shown in the figure. Point \(P\) is at a horizontal distance of 10 m from \(O\). If the pebble hits the block at \(t = 1~ s\), the value of \(v\) is \((\)take \(g = 10 ~m/s^2)\)


45 _————— o 10 m P 

Q.

A disc of radius R is cut out from a larger disc of radius 2R in such a way that the edge of the hole touches the edge of the disc.Locate the centre of mass of the residual disc.

Q. A body of moment of inertia of $3kg-m^2$ is rotating with an angular velocity of 2 rad/sec and has the same kinetic energy as a mass of 12 kg moving with a velocity of

1. 8 m/s
2. 0.5 m/s
3. 2 m/s
4. 1 m/s

Q. A block of mass $m$ is suspended from a pulley in the form of a circular disc of mass $m$ and radius $R$. The system is released from rest. Find the angular velocity of the disc when the block has dropped by height $h$. [Assume there is no slipping between string & pulley]

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

Q. In the figure shown, a ring $A$ is initially rolling without sliding with a velocity $v$ on a horizontal surface of the body $B$ (of same mass as $A$). All surfaces are smooth. $B$ has no initial velocity. What will be the maximum height reached by $A$ on $B$?

1. $\frac{v^2}{2g}$
2. $\frac{v^2}{3g}$
3. $\frac{3v^2}{4g}$
4. $\frac{v^2}{4g}$

Q. A thin hollow sphere of mass $m$ is completely filled with non-viscous liquid of mass $m$. When the sphere rolls on horizontal ground such that centre moves with velocity $v$, kinetic energy of the system is equal to :

1. $m{v}^2$
2. $\frac{4}{3}m{v}^2$
3. $\frac{4}{5}m{v}^2$
4. None of these

Q.

A body having a moment of inertia about its axis of rotation equal to 3 kg-m² is rotating with the angular velocity of 3 rad s⁻¹. The kinetic energy of this rotating body is the same as that of a body of mass 27 kg moving with velocity v. The value of v is-

1. 1 m/s

2. 1.5 m/s

3. 2.5 m/s

4. 2 m/s

Q.

No work is done when

1. a nail is plugged into a wooden board

2. a box is pushed along a horizontal floor

3. there is no component of force parallel to the direction of motion

4. there is no component of force normal to the direction of motion.

Q. The ratio of kinetic energy of two bodies is $2:1$ and their angular momentum are in the ratio of $1:2$. Then the ratio of their moment of inertia will be (assume pure rotational motion)

1. $1:4$
2. $4:1$
3. $8:1$
4. $1:8$

Q. A uniformly thick wheel, with moment of inertia $I$ and radius $R$, is free to rotate about its centre of mass (see fig.). A massless string is wrapped over its rim and two blocks of masses $m_1$ and $m_2>m_2$ are attached to the ends of the string. The system is released from rest. The angular speed of the wheel, when $m_1$ descents through a distance $h$, is

1. ${\left[\frac{2(m_1-m_2)gh}{(m_1+m_2)R^2+I}\right]}^{1/2}$
2. ${\left[\frac{2(m_1+m_2)gh}{(m_1+m_2)R^2+I}\right]}^{1/2}$
3. ${\left[\frac{(m_1-m_2)}{(m_1+m_2)R^2+I}\right]}^{1/2}gh$
4. ${\left[\frac{(m_1+m_2)}{(m_1+m_2)R^2+I}\right]}^{1/2}gh$

Q. Work done by a force on an object will be zero, if:

1. The force is always perpendicular to acceleration of object.
2. The object is at rest relative to ground but the point of application of force moves on the object.
3. The force is always perpendicular to velocity of object.
4. The point of application of force is fixed relative to ground but the object moves.

Q. A solid cylinder of mass ($M=1\text{kg}$) and radius $R=0.5\text{m}$ is pivoted at its centre The axis of rotation of the cylinder is horizontal. Three small particles of mass ($m=0.1\text{kg}$) are mounted along its surface as shown in figure. The system is initially at rest.

The angular speed of cylinder, when it has rotated through ${90}^{\circ }$ in anticlockwise direction.
(Take $g=10{\text{m/s}}^2$)

1. $\sqrt{5}\text{rad/s}$
2. $2\sqrt{3}\text{rad/s}$
3. $\sqrt{10}\text{rad/s}$
4. $4\sqrt{2}\text{rad/s}$

Q. Six small raindrops each of radius 1.5 mm, come down with a terminal velocity of $6cm{s}^{–1}.$ They coalesce to form a bigger drop. What is the terminal velocity (nearest integer in $m{s}^{–1})$ of the bigger drop?

Q. A disc of moment of inertia $\frac{9.8}{{\pi }^2}{\text{kg m}}^2$ is rotating at $600\text{rpm}$. If the frequency of rotation changes from $600\text{rpm}$ to $300\text{rpm}$, then what is the work done approximately?

1. $1467\text{J}$
2. $1452\text{J}$
3. $1567\text{J}$
4. $1632\text{J}$

Q. A ring of radius 0.5 m and mass 10 kg is rotating about its diameter with an angular velocity of 20 rad/s. Its kinetic energy is

1. 250 J
2. 500 J
3. 10 J
4. 100 J

Q. A solid cylinder is rolling without sliding. What fraction of its total kinetic energy is associated to rotational motion ?

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

Q. A thin rod $\text{MN}$, free to rotate in the vertical plane about the fixed end $\text{N}$, is held in a horizontal position. If the rod is released from this position, speed of end $\text{M}$ when the rod makes an angle $\alpha$ with the horizontal will be proportional to

1. $\sqrt{\cos \alpha }$
2. $\sin \alpha$
3. $\sqrt{\sin \alpha }$
4. $\cos \alpha$

Q. Upper half of an incline plane is rough and lower half is smooth. A cylinder starts its motion from the top. What is the ratio of its translational kinetic energy and rotational kinetic energy at the bottom ?

1. 2 : 1
2. 3 : 1
3. 4 : 1
4. 5 : 1

Q.

No work is done by a force on an object if

a) force is always perpendicular to its velocity

b) the object is stationary but the point of application of force moves on the object

c) the object moves in such a way that the point of application Of the force also moves

d) the force is always perpendicular to its acceleration.

Q. Three particles (A, B, C) each of mass $m$ are connected by three massless rods of length $l$. All three particles lie on smooth horizontal plane. A particle of mass $m$ moving along one of the rods with velocity $v_o$ strikes on a particle and stops (as shown in the diagram). Choose the correct option(s) among the following.

1. Tension in any of the rod after collision is $\frac{m{v}_o^2}{36l}$
2. Tension in any of the rod after collision is $\frac{m{v}_o^2}{28l}$
3. Loss of energy due to collision is $\frac{7m{v}_o^2}{12}$
4. Loss of energy due to collision is $\frac{7m{v}_o^2}{24}$

Q. In the graph shown below, find the amount of work done.

1. $0\text{J}$
2. $4\text{J}$
3. $2\text{J}$
4. $2\pi \text{J}$

Q. A ball falls on a wedge. The wedge is initially at rest. If the friction is absent everywhere and collision is elastic, then

1. mechanical energy as well as momentum is conserved in all direction.
2. mechanical energy is conserved but momentum is not conserved in all directions.
3. mechanical energy as well as momentum is not conserved.
4. mechanical energy is not conserved but momentum is conserved in all directions.

Q.

Work done by normal contact force on an object can be non-zero. Explain this statement.