Two persons of mass 55 kg and 65 kg respectively, are at the opposite ends of a boat.The length of the boat is 3.0m 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)3.0m                           

(2)2.3m

(3)zero                             

(4)0.75m

A solid cylinder of mass 3kg is rolling on a horizontal surface with velocity 4 $m{s}^{-1}.$ It collides with a horizontal spring of force constant 200 $N{m}^{-1}$. The maximum compression produced in the spring will be

1. 0.5 m

2. 0.6 m

3. 0.7 m

4. 0.2 m

A circular platform is mounted on a frictionless vertical axle. Its radius is R = 2m and its moment of inertia about the axle is 200 kg m². Initially, it is at rest. A 50 kg man stands on the edge of the platform and begins to walk along the edge at a speed of 1 m s^–1 relative to the ground. The time taken by the man to complete one revolution is:
1. $\mathrm{\pi } \sec$                 
2. $\frac{3\mathrm{\pi }}{2}\sec$
3. $2\mathrm{\pi } \sec$               
4. $\frac{\mathrm{\pi }}{2}\sec$

The moment of inertia of a uniform circular

disc is maximum about an axis perpendicular

to the disc and passing through

(1) B                   

(2) C

(3) D                   

(4) A

Three masses are placed on the x-axis:

300 g at origin, 500 g at x= 40 cm and 400g

at x=70 cm. The distance of the center of

mass from the origin is 

(1) 40 cm                   

(2) 45 cm

(3) 50 cm                   

(4) 30 cm 

The instantaneous angular position of a point on a rotating wheel is given by the equation \(\theta \left ( t \right )=2t^{3}-6t^{2}.\) The torque on the wheel becomes zero at:
1. \(t=0.5\) s
2. \(t=0.25\) s
3. \(t=2\) s
4. \(t=1\) s

A small mass attached to a string rotates on frictionless table top as shown. If the tension is the string is increased by pulling the string causing the radius of the circular motion to decrease by a factor of 2, the kinetic energy of the mass will

(a)remain constant

(b)increase by a factor of 2

(c)increase by a factor of 4

(d)decrease by a factor of 2

A circular disk of moment of inertia $I_t$ is rotating in a horizontal plane, about its symmetry axis, with a constant angular speed ${\omega }_i.$ Another disk of moment of inertia $I_b$ is dropped coaxially onto the rotation disk. Initially the second disk has zero angular speed. Eventually both the disks rotate with a constant angular speed ${\omega }_f.$ The energy lost by the initially rotating disc to friction is 

(1)$\frac{1}{2}\frac{I_b^2}{\left(I_t+I_b\right)}{\omega }_i^2$                                     

(2)$\frac{1}{2}\frac{I_t^2}{\left(I_t+I_b\right)}{\omega }_i^2$

(3)$\frac{1}{2}\frac{I_b-I_t}{\left(I_t+I_b\right)}{\omega }_i^2$                                     

(4)$\frac{1}{2}\frac{I_b{I}_t}{\left(I_t+I_b\right)}{\omega }_i^2$

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

1. 2v                                             2. 0

3. 1.5v                                          4. v