Point masses \(m_1\) and \(m_2\), are placed at the opposite ends of a rigid rod of length \(L\) and negligible mass. The rod is set into rotation about an axis perpendicular to it. The position of point \(P\) on this rod through which the axis should pass so that the ork required to set the rod rotating with angular velocity ω0 is minimum is given by:
     
1. \(x = \frac{m_1L}{m_1+m_2}\)
2. \(x= \frac{m_1}{m_2}L\)
3. \(x= \frac{m_2}{m_1}L\)
4. \(x = \frac{m_2L}{m_1+m_2}\)

Subtopic:  Moment of Inertia |
 72%
From NCERT
NEET - 2015
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A rod of weight \(w\) is supported by two parallel knife edges, A and B, and is in equilibrium in a horizontal position. The knives are at a distance \(d\) from each other. The centre of mass of the rod is at a distance \(x \) from A. The normal reaction on A is:

1. \(wx \over d\) 2. \(wd \over x\)
3. \(w(d-x) \over x\) 4. \(w(d-x) \over d\)
Subtopic:  Torque |
 66%
From NCERT
NEET - 2015
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A mass \(m\) moves in a circle on a smooth horizontal plane with velocity \(v_0\) at a radius \(R_0\). The mass is attached to a string that passes through a smooth hole in the plane, as shown.
   
The tension in the string is increased gradually and finally, \(m\) moves in a circle of radius \(\frac{R_0}{2}\). The final value of the kinetic energy is:
1. \( m v_0^2 \)
2. \( \frac{1}{4} m v_0^2 \)
3. \( 2 m v_0^2 \)
4. \( \frac{1}{2} m v_0^2\)

Subtopic:  Angular Momentum |
 58%
From NCERT
NEET - 2015
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Three identical spherical shells, each of mass \(m\) and radius \(r\) are placed as shown in the figure. Consider an axis \(XX'\), which is touching two shells and passing through the diameter of the third shell. The moment of inertia of the system consisting of these three spherical shells about the \(XX'\) axis is:

           
1. \(\frac{11}{5}mr^2\)
2. \(3mr^2\)
3. \(\frac{16}{5}mr^2\)
4. \(4mr^2\)

Subtopic:  Moment of Inertia |
 60%
From NCERT
NEET - 2015
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A force \(\vec{F}=\alpha \hat{i}+3 \hat{j}+6 \hat{k}\) is acting at a point  \(\vec{r}=2 \hat{i}-6 \hat{j}-12 \hat{k}\). The value of \(\alpha\) for which angular momentum about the origin is conserved is:
1. \(-1\)
2. \(2\)
3. zero
4. \(1\)

Subtopic:  Angular Momentum |
 68%
From NCERT
NEET - 2015
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An automobile moves on a road with a speed of \(54~\text{kmh}^{-1}.\)  The radius of its wheels is \(0.45\) m and the moment of inertia of the wheel about its axis of rotation is \(3~\text{kg-m}^2.\) If the vehicle is brought to rest in \(15\) s, the magnitude of average torque transmitted by its brakes to the wheel is:
1. \(6.66~\text{kg-m}^2\text{s}^{-2}\)
2. \(8.58~\text{kg-m}^2\text{s}^{-2}\)
3. \(10.86~\text{kg-m}^2\text{s}^{-2}\)
4. \(2.86~\text{kg-m}^2\text{s}^{-2}\)

Subtopic:  Torque |
 72%
From NCERT
NEET - 2015
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A solid cylinder of mass \(50\) kg and radius \(0.5\) m is free to rotate about the horizontal axis. A massless string is wound around the cylinder with one end attached to it and the other end hanging freely. The tension in the string required to produce an angular acceleration of \(2\) revolutions/s2 will be:
1. \(25\) N
2. \(50\) N
3. \(78.5\) N
4. \(157\) N

Subtopic:  Rotational Motion: Dynamics |
 52%
From NCERT
AIPMT - 2014
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A rod \(\mathrm{PQ}\) of mass \(M\) and length \(L\) is hinged at end \(\mathrm{P}\). The rod is kept horizontal by a massless string tied to point \(\mathrm{Q}\) as shown in the figure. When the string is cut, the initial angular acceleration of the rod is: 
            
1. \(\frac{g}{L}\)
2. \(\frac{2g}{L}\)
3. \(\frac{2g}{3L}\)
4. \(\frac{3g}{2L}\)

Subtopic:  Torque |
 79%
From NCERT
AIPMT - 2013
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The ratio of radii of gyration of a circular ring and a circular disc, of the same mass and radius, about an axis passing through their centers and perpendicular to their planes are:
1. \(1 : \sqrt 2\)
2. \(3:2\) 
3. \(2:1\)
4. \( \sqrt 2 : 1 \)
Subtopic:  Moment of Inertia |
 77%
From NCERT
NEET - 2013
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Two discs are rotating about their axes, normal to the discs and passing through the centres of the discs. Disc \(D_1\) has a \(2\) kg mass and \(0.2\) m radius and an initial angular velocity of \(50\) rads–1. Disc \(D_2\) has a \(4\) kg mass, \(0.1\) m radius and an initial angular velocity of \(200\) rad s–1. The two discs are brought in contact face to face, with their axes of rotation coincident. The final angular velocity (in rad s –1) of the system is:
1. \(60\)
2. \(100\)
3. \(120\)
4. \(40\)
Subtopic:  Angular Momentum |
 77%
From NCERT
NEET - 2013
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