11. Two persons of masses 55 kg and 65 kg respectively, are at the opposite ends of a boat. The length of the boat is 3.0 m and weighs 100 kg. The 55 kg man walks up to the 65 kg man and sits with him. If the boat is in still water the centre of mass of the system shifts by :

1. 1. 3.0 m

   
   
   

   2. 2.3 m

   
   
   

   3. zero

   
   
   

   4. 0.75 m

   
   
   

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1. View Hint View Answer Discuss in Forum

   
   There is no external force so centre of mass of the system will not shift

   Correct Option: C

   
   There is no external force so centre of mass of the system will not shift

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12. Three masses are placed on the x-axis : 300 g at origin, 500 g at x = 40 cm and 400 g at x = 70 cm. The distance of the centre of mass from the origin is :

1. 1. 40 cm

   
   
   

   2. 45 cm

   
   
   

   3. 50 cm

   
   
   

   4. 30 cm

   
   
   

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1. View Hint View Answer Discuss in Forum

   Xcm =	m1x1 + m2x2 + m3x3
   	m1 + m2 + m3

   
   

   Xcm =	300 × (0) + 500 × (40) + 400 × (70)
   	300 + 500 + 400

   
   

   Xcm =	500 × 40 + 400 × 70
   	1200

   
   

   Xcm =	50 + 70	=	120	= 40 cm
   	3		3

   Correct Option: A

   Xcm =	m1x1 + m2x2 + m3x3
   	m1 + m2 + m3

   
   

   Xcm =	300 × (0) + 500 × (40) + 400 × (70)
   	300 + 500 + 400

   
   

   Xcm =	500 × 40 + 400 × 70
   	1200

   
   

   Xcm =	50 + 70	=	120	= 40 cm
   	3		3

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13. Two particles which are initially at rest, move towards each other under the action of their internal attraction. If their speeds are v and 2v at any instant, then the speed of centre of mass of the system will be:

1. 1. 2v

   
   
   

   2. zero

   
   
   

   3. 1.5

   
   
   

   4. v

   
   
   

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1. View Hint View Answer Discuss in Forum

   If no external force actson a system of particles, the centre of mass remains at rest. So, speed of centre of mass is zero.

   Correct Option: B

   If no external force actson a system of particles, the centre of mass remains at rest. So, speed of centre of mass is zero.

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14. If the linear density (mass per unit length) of a rod of length 3m is proportional to x, where x is the distance from one end of the rod, the distance of the centre of gravity of the rod from this end is

1. 1. 2.5 m

   
   
   

   2. 1 m

   
   
   

   3. 1.5 m

   
   
   

   4. 2 m

   
   
   

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1. View Hint View Answer Discuss in Forum

   Consider an element of length dx at a distance x from end A.
   Here,  mass per unit length λ of rod
   λ ∝ x ⇒ λ = kx
   ∴ dm = λdx = kx dx
   
   Position of centre of gravity of rod from end A.
   

   Correct Option: D

   Consider an element of length dx at a distance x from end A.
   Here,  mass per unit length λ of rod
   λ ∝ x ⇒ λ = kx
   ∴ dm = λdx = kx dx
   
   Position of centre of gravity of rod from end A.
   

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15. A wheel has angular acceleration of 3.0 rad/sec² and an initial angular speed of 2.00 rad/sec. In a time of 2 sec it has rotated through an angle (in radian) of

1. 1. 10

   
   
   

   2. 12

   
   
   

   3. ​4

   
   
   

   4. 6

   
   
   

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1. View Hint View Answer Discuss in Forum

   Since, θ = ω0t +	1	= αt²
   	2

   
   Where α is angular aceleration, ω₀ is the initial angular speed.
   t = 2s
   

   θ = 2 × 2 +	1	× 3(2)² = 4 + 6 = 10 rad
   	2

   Correct Option: A

   Since, θ = ω0t +	1	= αt²
   	2

   
   Where α is angular aceleration, ω₀ is the initial angular speed.
   t = 2s
   

   θ = 2 × 2 +	1	× 3(2)² = 4 + 6 = 10 rad
   	2

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