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Three point masses $m_1$, $m_2$ and $m_3$ are placed at the corners of a thin massless
rectangular sheet (1.2 m x 1.0 m) as shown. Centre of mass will be located at the point 

1. (0.8, 0.6) m
2. (0.6, 0.8) m
3. (0.4, 0.4) m
4. (0.5,0.6) m

Figure shows a composite system of two uniform rods of lengths as indicated. Then the coordinates of the centre of mass of the system of rods are 

1. $\frac{L}{2},\frac{2L}{3}$
2. $\frac{L}{4},\frac{2L}{3}$
3. $\frac{L}{6},\frac{2L}{3}$
4. $\frac{L}{6},\frac{L}{3}$

A circular plate of diameter ‘a’ is kept in contact with a square plate of side a as
shown. The density of the material and the thickness are same everywhere. The
centre of mass of composite system will be

1. Inside the circular plate
2. Inside the square plate
3. At the point of contact
4. Outside the system

From a uniform square plate, one-fourth part is removed as shown. The
centre of mass of remaining part will lie on

1. OC
2. OA
3. OB
4. OD

A man of mass m is suspended in air by holding the rope of a balloon of mass
M. As the man climbs up the rope, the balloon
 

 
1. Moves upward
2. Moves downward
3. Remains stationary
4. Cannot stay

A man of mass m starts moving on a plank of mass M with constant velocity
v with respect to plank. If the plank lies on a smooth horizontal surface, then
velocity of plank with respect to ground is
1. $\frac{Mv}{m+M}$
2. $\frac{mv}{M}$
3. $\frac{Mv}{m}$
4. $\frac{mv}{m+M}$
 

Two particles A and B initially at rest move towards each other under a
mutual force of attraction. At the instant when velocity of A is v and that
of B is 2v, the velocity of centre of mass of the system
1. v
2. 2v
3. 3v
4. Zero

A shell following a parabolic path explodes somewhere in its flight. The
centre of mass of fragments will move in
1. Vertical direction
2. Any direction 
3. Horizontal direction
4. Same parabolic path

A ball of mass m is thrown upward and another ball of same mass is thrown
downward so as to move freely under gravity. The acceleration of centre of
mass is
1. g
2. g/2
3. 2g
4. Zero

The linear mass density($\lambda$) of a rod of length L kept along x-axis varies as $\lambda$ = $\alpha$
+ $\beta$x; where $\alpha$ and $\beta$ arepositive constants. The centre of mass of the rod is at
1. $\frac{\left(2\beta +3\alpha L\right)L}{2\left(2\beta +\alpha L\right)}$
2. $\frac{\left(3\alpha +2\beta L\right)L}{3\left(2\alpha +\beta L\right)}$
3. $\frac{\left(3\beta +2\alpha L\right)L}{3\left(2\beta +\alpha L\right)}$
4. $\frac{\left(3\beta +2\alpha L\right)L}{3\beta +2\alpha }$

A man of mass 60 kg is standing on a boat of mass 140 kg, which is at rest in
still water. The man is initially at 20 m from the shore. He starts walking on the
boat for 4 s with constant speed 1.5 rn/s towards the shore. The final distance of
the man from the shore is 
1. 15.8 m
2. 4.2 m
3. 12.6 m
4. 14.1 m

A bomb of mass m is projected from the ground with speed y at angle $\theta$ with the
horizontal. At the maximum height from the ground it explodes into two
fragments of equal mass. If one fragment comes to rest immediately after explosion, then the horizontal range of centre of mass is
1. $\frac{v^2{\sin }^2\theta }{g}$
2. $\frac{v^2{\sin }^{}\theta }{g}$
3. $\frac{v^2{\sin }^{}\theta }{2g}$
4. $\frac{v^2{\sin }^{}2\theta }{g}$

Two blocks of masses 5 kg and 2 kg are connected by a spring of negligible mass
and placed on a frictionless horizontal surface. An impulse gives a velocity of 7
rn/s to the heavier block in the direction of the lighter block. The velocity of the
centre of mass is
1. 30 m/s
2. 20 m/s
3. 10 m/s
4. 5 m/s