A particle is moved along a path AB-BC-CD-DE-EF-FA, as shown in figure, in presence of a force $$F = (\alpha y\widehat i + 2\alpha x\widehat j)$$ N, where x and y are in meter and $$\alpha $$ = $$ - $$1 Nm^-1. The work done on the particle by this force F will be ............... Joule.

A spring-block system is resting on a frictionless floor as shown in the figure. The spring constant is $$2.0\,N{m^{ - 1}}$$ and the mass of the block is $$2.0$$ $$kg.$$ Ignore the mass of the spring. Initially the spring is an unstretched condition. Another block of mass $$1.0$$ $$kg$$ moving with a speed of $$2.0$$ $$m{s^{ - 1}}$$ collides elastically with the first block. The collision is such that the $$2.0$$ $$kg$$ block does not hit the wall. The distance, in metres, between the two blocks when the spring returns to its unstretched position for the first time after the collision is __________.

Consider an elliptically shaped rail PQ in the vertical plane with OP = 3 m and OQ = 4 m. A block of mass 1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is (n × 10) Joules. The value of n is (take acceleration due to gravity = 10 ms^–2)

A bob of mass m, suspended by a string of length l₁ is given a minimum velocity required to complete a full circle in the vertical plane. At the highest point, it collides elastically with another bob of mass m suspended by a string of length l₂, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio $${{{l_1}} \over {{l_2}}}$$ is