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Q. No.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 |
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A thin uniform hemispherical shell (radius: \(R,\) center: \(O\)) is cut into two symmetric quarter spheres by means of a vertical plane, as shown. The centre-of-mass of a quarter sphere is at a distance \(d\) from \(O\). Then \(d\)=
| 1. | \(\dfrac R2\) | 2. | \(\dfrac R{\sqrt2}\) |
| 3. | \(\dfrac R{4}\) | 4. | \(\dfrac R{2\sqrt2}\) |
Subtopic: Â Center of Mass |
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A uniform solid hemisphere of mass \(m\) and radius \(R\) is rotated about an axis passing through its center \(O\) along a diameter of its flat surface. The moment of inertia of the hemisphere, about this axis, is:
1. \(\dfrac15mR^2\)
2. \(\dfrac25mR^2\)
3. \(\dfrac13mR^2\)
4. \(\dfrac23mR^2\)
1. \(\dfrac15mR^2\)
2. \(\dfrac25mR^2\)
3. \(\dfrac13mR^2\)
4. \(\dfrac23mR^2\)
Subtopic: Â Moment of Inertia |
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A uniform rod of mass \(M\) and length \(L\) lies at rest on a smooth horizontal plane, as shown in the figure. A particle of mass \(m,\) moving with an initial velocity \(u\) strikes one end \((A)\) of the rod and stops. The initial velocity \(u\) is perpendicular to the length \((AB)\) of the rod.
Consider the following statements:
Choose the correct option from the given ones:
Consider the following statements:
| (P) | The momentum of the system is conserved. |
| (Q) | The kinetic energy of the system does not change after the collision. |
| (R) | The angular momentum of the system is conserved. |
Choose the correct option from the given ones:
| 1. | P, Q, and R are true. |
| 2. | P and R are true. |
| 3. | Only R is true. |
| 4. | Only P is true. |
Subtopic: Â Angular Momentum |
 52%
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A block of mass \(m\) slides down the smooth inclined surface of a wedge of mass \(M\) starting from rest. The wedge is at rest on the horizontal surface beneath it, due to friction. The acceleration of the center of mass of the system of the block and the wedge is:
| 1. | \(\dfrac{mg~\text{sin}\theta}{m+M}\) | 2. | \(\dfrac{mg~\text{cos}\theta}{m+M}\) |
| 3. | \(g~\text{sin}\theta\) | 4. | zero |
Subtopic: Â Center of Mass |
 52%
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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 |
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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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