System of Parti...

Three particles of masses 1 kg, 2 kg and 3 kg are situated at the corners of the equilateral triangle move at speed 6 ms$$^{-1}$$, 3 ms$$^{-1}$$ and 2 ms$$^{-1}$$ respectively. Each particle maintains a direction towards the particle at the next corner symmetrically. Find velocity of COM of the system at this instant :

A streamlined and almost symmetrical car of mass $$M$$ has centre of gravity at a distance $$p$$ from the rear wheel, $$q$$ from the front wheel and $$h$$ from the road. If the car has required power and friction, the maximum acceleration developed without tipping over towards back is:

A steel plate of thickness $$h$$ has the shape of a square whose side equals $$l$$, with $$h\ll l$$. The plate is rigidly fixed to a vertical axle $$OO$$ which is rotated with a constant angular acceleration $$\beta$$ (figure shown above). Find the deflection $$\lambda$$ assuming the sagging to be small.

A particle of mass m comes down on a smooth inclined plane from point B at a height of h from rest. The magnitude of change in momentum of the particle between position A (just before arriving on horizontal surface) and C (assuming the angle of inclination of the plane as $$\theta$$ with respect to the horizontal) is :

The radius of gyration of a plane lamina of mass $$M$$, length $$L$$ and breadth $$B$$ about an axis passing through its center of gravity and perpendicular to its plane will be

Four identical rods are joined end to end to form a square. The mass of each rod is $$M$$. The moment of inertia of the square about the median line is 

Two particles of equal mass have initial velocities $$2\hat{i}\, ms^{-1}\, and\, 2\hat{j}\, ms^{-1}$$. First particle has a constant acceleration $$(\hat{i}\,+\,\hat{j})\, ms^{-1}$$ while the acceleration of the second particle is always zero. The centre of mass of the two particles moves in :

Three rings each of mass m and radius r are so placed that they touch each other. The radius or gyration of the system about the axis as shown in the figure is:

Consider the following two statement-
[A] Linear momentum of a system of particle is zero
[B] kinetic energy of a system of particles is zero. Then

Linear mass density of the two rods system, $$AC$$ and $$CB$$ is $$x$$. Moment of inertia of two rods about an axis passing through $$AB$$ is