1. A uniform rigid rod of mass M and length L is standing vertically along the y-axis on a smooth horizontal surface, with its lower end at the origin (0,0). A slight disturbance at t = 0 causes the lower end to slip on the smooth surface along the positive x-axis, and the rod starts falling.
What is the path followed by the centre of mass of the rod during its fall?
2. A hemispherical bowl of radius R = 0.1 m is rotating about its own axis (which is vertical), with an angular velocity ω. A particle of mass 10-2 kg on the frictionless inner surface of the bowl is also rotating with the same ω. The particle is at a height h from the bottom of the bowl.
Obtain the relation between h and ω. What is minimum value of ω needed, in order to have a non-zero value of h?
3. A cylindrical solid of mass 10-2 and cross-sectional area 10-4 m2 is moving parallel to its axis (the x-axis) with a uniform speed of 103 m/s in the positive direction. At t = 0, its front face passes the dust particles of uniform density 10-3 kg/m3. When a dust particle collides with the face of the cylinder, it sticks to its surface. Assuming that the dimensions of the cylinder remain practically unchanged, and that the dimensions of the cylinder remain practically unchanged, and that the dust sticks only to the front face of the cylinder, find the x-coordinate of the front of the cylinder at t = 150 s.
4. A circular ring of radius R with uniform positive charge density λ per unit length is located in the y-z plane with its centre at the origin O. A particle of mass m and positive charge q is projected from the point P(R√3,0,0) on the positive x-axis directly towards O, with an initial speed v. Find the smallest (non-zero) value of the speed v such that the particle does not return to P.
5. Two radio stations broadcast their programmes at the same amplitude A, and at slightly different frequencies ω1 and ω2 respectively, where ω2 - ω1 = 10³ Hz. A detector receives the signals from the two stations simulataneously. It can only detect signals of intensity ≥2A². Find the time interval between successive maxima of the intensity of the signal received by the detector.
Source (1993 JEE paper)