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Q. No.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)
(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
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.
Subtopic: Â Center of Mass |
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A horizontal force \(F\) is applied to a uniform solid sphere at rest, so that its line of action passes through the mid-point (\(P\)) of the vertical radius \(OA;O\) being the centre of the sphere (mass : \(m\)). The acceleration of the uppermost point \(A\) is:
| 1. | equal to \(\dfrac{F}{m}.\) |
| 2. | greater than \(\dfrac{F}{m}.\) |
| 3. | less than \(\dfrac{F}{m}.\) |
| 4. | unpredictable, and depends on the radius of the sphere. |
Subtopic: Â Torque |
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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 |
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A thin uniform hemispherical shell (radius: \(R\)) is placed upside down: with its open end down. It is cut into two parts by a horizontal plane, so that the masses of the two parts are equal. The centre-of-mass of the lower part is located at a distance of:
1. \(\dfrac R3\) from \(O\)
2. \(\dfrac R4\) from \(O\)
3. \(\dfrac R6\) from \(O\)
4. \(\dfrac R8\) from \(O\)
1. \(\dfrac R3\) from \(O\)
2. \(\dfrac R4\) from \(O\)
3. \(\dfrac R6\) from \(O\)
4. \(\dfrac R8\) from \(O\)
Subtopic: Â Center of Mass |
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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 |
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The centre-of-mass of a uniform solid cone is located at a distance of \(\dfrac{3}{4} H\) from the vertex of the cone, where \(H\) is its height. The centre-of-mass of a regular solid tetrahedron of height \(H\) is located at a distance of:
1. \(\dfrac{1}{2} H\) from its vertex
2. \(\dfrac{2}{3}H\) from its vertex
3. \(\dfrac{3}{4}H\) from its vertex
4. \(\dfrac{3}{8}H\) from its vertex
1. \(\dfrac{1}{2} H\) from its vertex
2. \(\dfrac{2}{3}H\) from its vertex
3. \(\dfrac{3}{4}H\) from its vertex
4. \(\dfrac{3}{8}H\) from its vertex
Subtopic: Â Center of Mass |
 52%
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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 |
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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 |
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The centre-of-mass of a uniform solid hemisphere of radius \(R\) is located at a distance of \(\dfrac{3R}{8}\) from its centre \(O.\) If the hemisphere is now cut symmetrically into two identical parts by a single plane, the centre-of-mass of one of the parts is located at a distance from \(O\) of:
1. \(\dfrac{3R}{8}\)
2. \(\dfrac{3R}{4}\)
3. \(\dfrac{3R}{8}\sqrt2\)
4. \(\dfrac{3R}{5}\sqrt2\)
1. \(\dfrac{3R}{8}\)
2. \(\dfrac{3R}{4}\)
3. \(\dfrac{3R}{8}\sqrt2\)
4. \(\dfrac{3R}{5}\sqrt2\)
Subtopic: Â Center of Mass |
 54%
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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 |
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