Q: Two particular 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 seperation between them becomes 2x is

(a) $\large \frac{F}{2m}\frac{a}{\sqrt{a^2 – x^2}}$

(b) $\large \frac{F}{2m}\frac{x}{\sqrt{a^2 – x^2}}$

(c) $\large \frac{F}{2m}\frac{x}{a}$

(d) $\large \frac{F}{2m}\frac{\sqrt{a^2 – x^2}}{x}$

Ans: (b)

Sol: When string is pulled by a force F , let tension created is T

$\large 2 T cos\theta = F$

$\large T = \frac{F}{2}sec\theta$

Acceleration of the particle is

$\large a = \frac{T sin\theta}{m}$

$\large a = \frac{F}{2m}tan\theta$

$\large = \frac{F}{2m}\frac{x}{\sqrt{a^2-x^2}}$