Systems of Part...

A sphere is rolling without slipping on a fixed horizontal plane surface. In the figure, A is the point of contact, B is the centre of the sphere and C is its topmost point. Then,

A solid cylinder is rolling down a rough inclined plane of inclination θ. Then

The moment of inertia of thin square plate ABCD of uniform thickness about an axis passing through the centre O and perpendicular to the plane of the plate is
where I₁, I₂, I₃ and I₄ are respectively moments of inertia about axes 1, 2, 3 and 4 which are in the plane of the plate.

A uniform bar of length 6a and mass 8 m lies on a smooth horizontal table. Two point masses m and 2 m moving in the same horizontal with speed 2v and v respectively, strike the bar (as shown in the fig.) and stick to the bar after collision. Denoting angular velocity (about the centre of mass), total energy and centre of mass velocity by ω, E and v_C respectively, we have after collision

A particle of mass m is projected with a velocity v making an angle of 45° with the horizontal. The magnitude of the angular momentum of the projectile about the point of projection, when the particle is at its maximum height h, is

A solid cylinder of mass m and radius r is rolling on a rough inclined plane of inclination θ. The coefficient of friction between the cylinder and incline is µ. Then

The figure shows a system consisting of (i) ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed ω and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed ω/2. The ring and disc are separated by frictionless ball bearings. The system is in the x–z plane. The point P on the inner disc is at a distance R from the origin, where OP makes an angle of 30° with the horizontal. Then with respect to the horizontal surface

Statement I If there is no external torque on a body about its centre of mass, then the velocity of the centre of mass, remains constant.
Statement II The linear momentum of an isolated system remains constant.

Statement I Two cylinders, one hollow (metal) and the other solid (wood) with the same mass and identical dimensions are simultaneously allowed to roll without slipping down inclined plane from the same height. The hollow cylinder will reach the bottom of the inclined plane first.
Statement II By the principle of conservation of energy, the total kinetic energies of both the cylinders are identical when they reach the bottom of the incline.