A roller is made by joining together two cones at their vertices $$0$$. It is kept on two rails $$AB$$ and $$CD$$, which are placed asymmetrically (see figure), with its axis perpendicular to $$CD$$ and its center $$O$$ at the center of line joining $$AB$$ and $$CD$$ (see figure). It is given a light push so that it starts rolling with its center $$O$$ moving parallel to $$CD$$ in the direction shown. As it moves, the roller will tend to :

A pendulum clock loses $$12$$ $$s$$ a day if the temperature is $${40^ \circ }C$$ and gains $$4$$ $$s$$ a day if the temperature is $${20^ \circ }C.$$ The temperature at which the clock will show correct time, and the co-efficient of linear expansion $$\left( \alpha \right)$$ of the metal of the pendulum shaft are respectively :

An ideal gas undergoes a quasi static, reversible process in which its molar heat capacity $$C$$ remains constant. If during this process the relation of pressure $$P$$ and volume $$V$$ is given by $$P{V^n} = $$ constant, then $$n$$ is given by (Here $${C_p}$$ and $${C_v}$$ are molar specific heat at constant pressure and constant volume, respectively:

A point particle of mass $$m,$$ moves long the uniformly rough track $$PQR$$ as shown in the figure. The coefficient of friction, between the particle and the rough track equals $$\mu .$$ The particle is released, from rest from the point $$P$$ and it comes to rest at point $$R.$$ The energies, lost by the ball, over the parts, $$PQ$$ and $$QR$$, of the track, are equal to each other , and no energy is lost when particle changes direction from $$PQ$$ to $$QR$$.

The value of the coefficient of friction $$\mu $$ and the distance $$x$$ $$(=QR),$$ are, respectively close to: