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:

When a rubber-band is stretched by a distance $$x$$, it exerts restoring force of magnitude $$F = ax + b{x^2}$$ where $$a$$ and $$b$$ are constants. The work done in stretching the unstretched rubber-band by $$L$$ is :

This question has Statement $$1$$ and Statement $$2.$$ Of the four choices given after the Statements, choose the one that best describes the two Statements.

If two springs $${S_1}$$ and $${S_2}$$ of force constants $${k_1}$$ and $${k_2}$$, respectively, are stretched by the same force, it is found that more work is done on spring $${S_1}$$ than on spring $${S_2}$$.

STATEMENT 1: If stretched by the same amount work done on $${S_1}$$, Work done on $${S_1}$$ is more than $${S_2}$$
STATEMENT 2: $${k_1} < {k_2}$$

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $$U\left( x \right) = {a \over {{x^{12}}}} - {b \over {{x^6}}},$$ where $$a$$ and $$b$$ are constants and $$x$$ is the distance between the atoms. If the dissociation energy of the molecule is $$D = \left[ {U\left( {x = \infty } \right) - {U_{at\,\,equilibrium}}} \right],\,\,D$$ is