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Q. No.A block of mass \(m\) slides down the smooth inclined surface of a wedge of mass \(M;\) which is itself on a smooth horizontal surface. The centre-of-mass of the system:
| 1. | is stationary |
| 2. | accelerates to the left |
| 3. | accelerates to the right |
| 4. | accelerates downward |
Subtopic: Â Center of Mass |
Level 4: Below 35%
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A uniform ladder of mass \(10\) kg is placed at an angle against a frictionless vertical wall, as shown in the figure, by applying a horizontal force \(F\) at the bottom \((B)\) of the ladder, towards the wall. (Take \(g=10\) m/s2).
Assume that the ground is frictionless. The force \(F\) equals:
Assume that the ground is frictionless. The force \(F\) equals:
| 1. | \(100\sqrt3\) N | 2. | \(50\sqrt3\) N |
| 3. | \(\dfrac{100}{\sqrt3}\) N | 4. | \(\dfrac{50}{\sqrt3}\) N |
Subtopic: Â Torque |
Level 3: 35%-60%
Hints
A uniform chain \(ABC\) is at rest with half of it hanging off the edge of a rough horizontal table. If the total length of the chain is \(2~\text m,\) the centre-of-mass of the chain is located:
| 1. | \(0.25~\text{m}\) below the table |
| 2. | \(0.5~\text{m}\) below the table |
| 3. | \(0.33~\text{m}\) below the table |
| 4. | \(0.4~\text{m}\) below the table |
Subtopic: Â Center of Mass |
Level 3: 35%-60%
Hints
A block of mass \(m\) is placed atop another block of mass \(M,\) and the combination is at rest on a smooth horizontal table. A force \(F_1\) is applied to \(m\) and another force \(F_2\) is applied to \(M,\) the two acting horizontally and in opposite directions. Consider the following statements about the acceleration \((a_{cm})\) of the centre of mass of the system).
(take right as positive)
Choose the most appropriate option from the given ones:
1. only (A) is True.
2. only (B) is True.
3. (C) is True.
4. (A) and (B) are True but (C) is False.
(take right as positive)
| (A) | \(a_{cm}=\dfrac{F_1-F_2}{m+M},\) if there is no friction acting between \(m\) and \(M\) |
| (B) | \(a_{cm}=\dfrac{F_1-F_2}{m+M},\) if there is static friction between \(m\) and \(M\) |
| (C) | \(a_{cm}=\dfrac{F_1-F_2}{m+M},\) in all situations |
1. only (A) is True.
2. only (B) is True.
3. (C) is True.
4. (A) and (B) are True but (C) is False.
Subtopic: Â Center of Mass |
Level 3: 35%-60%
Hints
A block \(A\) is pushed on a smooth horizontal plane by applying a horizontal force \(F ,\) which causes an acceleration of \({\dfrac g 4}\) (\(g\): acceleration due to gravity). The block does not topple, even though the force acts at its highest point. The normal reaction shifts forward by:
| 1. | \({\dfrac b 2}\) | 2. | \({ \dfrac b 4}\) |
| 3. | \({\dfrac b 8}\) | 4. | \(\dfrac b 3\) |
Subtopic: Â Torque |
Level 3: 35%-60%
Hints
The moment of inertia of a uniform right-angled triangular lamina (mass: \(m\)) \(\Delta ABC \) about an axis passing through \(C,\) perpendicular to its plane is:
| 1. | \(m\left(\dfrac{a^2 +b^2}{3}\right ) \) | 2. | \(m\left(\dfrac{a^2 +b^2}{6}\right) \) |
| 3. | \(m\left(\dfrac{a^2 +b^2}{12}\right) \) | 4. | \(m\left(\dfrac{a^2 +b^2}{2}\right) \) |
Subtopic: Â Moment of Inertia |
Level 3: 35%-60%
Hints
The moment of inertia of the uniform rod of mass \(m,\) length \(L\) about the axis shown in the figure is \(\dfrac14mL^2.\) Then, the angle \(\theta\) is:
1. \(\text{sin}^{-1}\left(\dfrac34\right) \)
2. \(\text{tan}^{-1}\left(\dfrac34\right) \)
3. \(60^{\circ}\)
4. \(30^{\circ}\)
1. \(\text{sin}^{-1}\left(\dfrac34\right) \)
2. \(\text{tan}^{-1}\left(\dfrac34\right) \)
3. \(60^{\circ}\)
4. \(30^{\circ}\)
Subtopic: Â Moment of Inertia |
 50%
Level 3: 35%-60%
Hints
A uniform rod of length \(L\) is standing upright, pivoted at its lower end. The rod can freely rotate about the pivot. If it is slightly disturbed so that it falls to the ground, the speed of the highest point, when it strikes the ground will be:
| 1. | \(\sqrt{2gL}\) | 2. | \(\sqrt{3gL}\) |
| 3. | \(\sqrt{6gL}\) | 4. | \(\sqrt{gL}\) |
Subtopic: Â Rotational Motion: Dynamics |
Level 3: 35%-60%
Hints
A thin spherical metallic vessel of radius \(R\) contains water, the mass of water being equal to the mass of the vessel that contains it. A hole is made in the bottom so that the water begins to flow out. When the vessel is half-empty the centre-of-mass is at a distance \(d\) from the centre of the vessel:
| 1. | \(d=\dfrac{3 R}{16}\) | 2. | \(d=\dfrac{R}{2}\) |
| 3. | \(d=\dfrac{R}{4}\) | 4. | \(d=\dfrac{R}{8}\) |
Subtopic: Â Center of Mass |
Level 3: 35%-60%
Hints
A uniform cylinder of mass \(M,\) radius \(R\) and height \(3R\) is placed upright on a horizontal surface. A particle of mass \(m\) is placed on the top of the cylinder at its edge. For what minimum value of \(m\) will the cylinder topple?
| 1. | \(m = 3M\) |
| 2. | \(m= \dfrac {M}{3}\) |
| 3. | \(m= \dfrac {3M }{2}\) |
| 4. | No value of \(m\) will cause the cylinder to topple. |
Subtopic: Â Torque |
Level 3: 35%-60%
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