Two particles of mass m each are tied at the ends of a light string of length 2a. The whole system is kept on a frictionless horizontal surface with the string held tight so that each mass is at a distance `a` from the centre P (as shown in the figure). Now, the mid-point of the string is pulled vertically upwards with a small but constant force F. As a result, the particles move towards each other on the surface. The magnitude of acceleration, when the separation between them becomes 2x, is

Explanation

Let the tension in the string be T at any angular position `theta`. The acceleration of each ball along x and y axes be a and `a_(1)`, respectively. Writing the equation of motion of m, we obtain
`Sigma F_(x)=ma implies T cos theta =ma` ..(i)
`Sigma F_(y)=ma_(1) implies T sin theta =ma_(1)` ...(ii)
At point P, as it is accelerating with an acceleration a, therefore `F-2t cos theta =m_(P)a`
Where `m_(p)`=mass of the string at the point `P~=0`
`implies F=2T cos theta` ..(iii)
`(ii) -: (iii) implies tan theta =(a_(1))/(a)`
`implies a_(1)=a tan theta`, where `a=(T cos theta)/(m)` from (i).
Putting `T=(F)/(2 cos theta)` from (3), we obtain
`a_(1)=(F)/(2m) tan theta`.