In given pulley mass system, mass m₁ = 500 g, m₂ = 460 g and the pulley has a radius of 5 cm. When released from rest, heavier mass falls through 7.50 cm in 5 s. There is no slippage between pulley and string.
(i) What is magnitude of acceleration of mass?
(ii) What is magnitude of pulley's angular acceleration?

Find the centre of mass for a solid cone of base radius r and height h.

A man stands on a rotating platform with his arms stretched horizontally holding a 5 kg weight in each hand. The angular speed of the platform in 30 rpm. The man then brings his arms close to his body with the distance of each weight from the axis changing from 90 cm to 20 cm. The moment of inertia of the man together with the platform may be taken to be constant and equal to 7.6 kg-m².
Is kinetic energy conserved in the process? If not, from where does the change come about?

A cylinder is released from rest from the top of an incline of inclination \(\theta \) and length l. If the cylinder rolls without slipping. What will be its speed when it reaches the bottom?

Show that a.(b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors , a, b and c.

Find the components along the x, y, z axes of the angular momentum l of a particle, whose position vector is r with components x, y, z and momentum is p with components P_x,P_y and P_z. Show that if the particle moves only in the x-y plane the angular momentum has only a z-component.

From a uniform disk of radius R, a circular hole of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of gravity of the resulting flat body.

A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination.
(a) Will it reach the bottom with the same speed in each case?
(b) Will it take longer to roll down one plane than the other?
(c) If so, which one and why?

Prove the theorem of parallel axes.[Hint: If the centre of mass of chosen the origin\(\sum { { m }_{ i }r_{ i } } =0\)]

Discuss the rolling motion of a cylinder on an inclined plane.