A football of radius R is kept on a hole of radius r (r < R) made on a plank kept horizontally. One end of the plank is now lifted so that it gets tilted making an angle $$\theta $$ from the horizontal as shown in the figure below. The maximum value of $$\theta $$ so that the football does not start rolling down the plank satisfies (figure is schematic and not drawn to scale)

A wire, which passes through the hole in a small bead, is bent in the form of quarter of a circle. The wire is fixed vertically on ground as shown in the below figure. The bead is released from near the top of the wire and it slides along the wire without friction. As the bead moves from A to B, the force it applies on the wire is

A block of mass m₁ = 1 kg another mass m₂ = 2 kg, are placed together (see figure) on an inclined plane with angle of inclination $$\theta$$. Various values of $$\theta$$ are given in List I. The coefficient of friction between the block m₁ and the plane is always zero. The coefficient of static and dynamic friction between the block m₂ and the plane are equal to $$\mu$$ = 0.3. In List II expressions for the friction on the block m₂ are given. Match the correct expression of the friction in List II with the angles given in List I, and choose the correct option. The acceleration due to gravity is denoted by g.

[Useful information : tan (5.5$$^\circ$$) $$\approx$$ 0.1; tan (11.5$$^\circ$$) $$\approx$$ 0.2; tan (16.5$$^\circ$$) $$\approx$$ 0.3]

List I		List II
P.	$$\theta = 5^\circ $$	1.	$${m_2}g\sin \theta $$
Q.	$$\theta = 10^\circ $$	2.	$$({m_1} + {m_2})g\sin \theta $$
R.	$$\theta = 15^\circ $$	3.	$$\mu {m_2}g\cos \theta $$
S.	$$\theta = 20^\circ $$	4.	$$\mu ({m_1} + {m_2})g\cos \theta $$

A ball of mass (m) 0.5 kg is attached to the end of a string having length (L) 0.5 m. The ball is rotated on a horizontal circular path about vertical axis. The maximum tension that the string can bear is 324 N. The maximum possible value of angular velocity of ball (in radian/s) is