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A uniform chain of length \(L\) is held so that its lowest point is in contact with the ground. The chain is released. When half of the chain has fallen, the centre-of-mass has descended by:
                        
1. \(\large\dfrac{L}{2}\) 2. \(\large\dfrac{L}{4}\)
3. \(\large\dfrac{L}{8}\) 4. \(\large\dfrac{3L}{8}\)
Subtopic:  Center of Mass |
Level 4: Below 35%

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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:  Center of Mass |
Level 4: Below 35%

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Given below are two statements: 
Assertion (A): The centre-of-mass of a solid body lies within the body.
Reason (R): The centre-of-mass of any two particles lies between them, on the straight line segment joining them.
 
1. Both (A) and (R) are True and (R) is the correct explanation of (A).
2. Both (A) and (R) are True but (R) is not the correct explanation of (A).
3. (A) is True but (R) is False.
4. (A) is False but (R) is True.
Subtopic:  Center of Mass |
Level 4: Below 35%

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Two identical projectiles are simultaneously launched into the air, with the same speed but in opposite directions – inclined at \(60^\circ\) to the horizontal. Ignore any air resistance. Their centre-of-mass:
1. remains at rest
2. moves along a horizontal straight line
3. moves along a vertical straight line
4. moves along a parabolic path
Subtopic:  Center of Mass |
Level 4: Below 35%

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A uniform square sheet of metal \((ABCD)\) has a triangular piece \(OCD\) cut out of it, where \(O\) is the centre of the original square. The side of the original square is \(a.\) The distance, of the centre-of-mass of the remaining portion, from the centre \(O\) is:
           
1. \(\dfrac{a}{6}\) 2. \(\dfrac{a}{12}\)
3. \(\dfrac{a}{9}\) 4. \(\dfrac{a}{4}\)
Subtopic:  Center of Mass |
Level 3: 35%-60%

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The moment of inertia of a uniform square lamina, of mass \(m\), about one of its diagonals (diagonal-length: \(d\)) is:
1. \(\dfrac{md^2}{6}\)

2. \(\dfrac{md^2}{3}\)

3. \(\dfrac{md^2}{12}\)

4. \(\dfrac{md^2}{24}\)

Subtopic:  Moment of Inertia |
Level 3: 35%-60%

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Two identical uniform semi-circular pieces are joined smoothly at their ends so as to form a planar \(S\)-shape. The mass of each piece is \(m\) and radius \(R.\) The moment of inertia of this \(S,\) about the common tangent at its joint (at \(P\)) is given by:
   
1. \(mR^2\) 2. \({\dfrac32}mR^2\)
3. \(2mR^2\) 4. \(3mR^2\)
Subtopic:  Moment of Inertia |
Level 3: 35%-60%

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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:
1. \(100\sqrt3\) N 2. \(50\sqrt3\) N
3. \(\dfrac{100}{\sqrt3}\) N 4. \(\dfrac{50}{\sqrt3}\) N
Subtopic:  Torque |
Level 3: 35%-60%

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A uniform wheel (of mass \(m\)) rotates about its axis so that the speed of a point on its rim is \(v.\) The angular momentum of the wheel, about its axis, is:
1. \(mvr\) 2. \({\dfrac12}mvr\)
3. \({\dfrac32}mvr\) 4. \({\dfrac14}mvr\)
Subtopic:  Angular Momentum |
Level 3: 35%-60%

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The centre-of-mass (C.M.) of four identical particles kept at the four vertices of a square with a diagonal \(d,\) is at its geometric centre. If one of the particles is removed, the C.M. of the remaining particles is located at a distance, from the centre, of:
1. \(\dfrac{d}{4}\) 2. \(\dfrac{d}{3}\)
3. \(\dfrac{d}{6}\) 4. \(\dfrac{3d}{4}\)
Subtopic:  Center of Mass |
Level 3: 35%-60%

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