two particles each of mass m are tied at the ends of a light string of length 2a. the whole system is kept on a frictionless horizontal surface with a string held tight so that each mass is 'a' distance away frm the centre P Now the midpoint of the string is pulled vertically upwards with a small but constant force F. As a result, the particals move towards each other on the surface. Find the magnitude of acceleration, when the separation betweenthem becomes 2x.

To solve this problem, we need to analyze the motion of the two particles when the midpoint of the string is pulled upwards. Initially, the two particles are positioned at a distance of 'a' from the center point P, making the total length of the string 2a. When the midpoint is pulled upwards with a force F, the particles will move towards each other due to the tension created in the string. Let's break this down step by step.

Understanding the Setup

We have two particles, each with mass m, connected by a light string of length 2a. The midpoint of the string is at point P, and when the string is pulled upwards, the particles will move closer together on a frictionless surface. The goal is to find the acceleration of the particles when their separation becomes 2x.

Analyzing Forces and Motion

When the midpoint of the string is pulled upwards, a tension T is created in the string. This tension acts horizontally on both particles, causing them to accelerate towards each other. The key here is to understand how the vertical force translates into horizontal motion.

Applying Newton's Second Law

According to Newton's second law, the net force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). In our case, the tension T in the string will be the net force acting on each particle. Therefore, we can write:

• For each particle: T = m * a

Here, 'a' is the acceleration of each particle towards the center. Since both particles are experiencing the same tension, they will have the same acceleration.

Relating Tension to the Force F

The vertical force F applied at the midpoint creates a tension T in the string. The relationship between the vertical force and the tension can be derived from the geometry of the system. When the midpoint is pulled up, the angle θ formed by the string with the horizontal can be approximated for small displacements. The vertical component of the force F contributes to the tension in the string.

Using trigonometry, we can express the tension as:

• T = F / sin(θ)

For small angles, sin(θ) can be approximated as θ (in radians), which is equal to the vertical displacement divided by the horizontal distance (which is a - x). Thus, we can express T in terms of F and the geometry of the system.

Finding the Acceleration

Now, when the separation between the particles becomes 2x, the distance between them is reduced from 2a to 2x. The horizontal distance each particle moves towards the center is (a - x). The relationship between the tension and the force F gives us:

• T = F * (a - x) / (a - x)

Substituting this into the equation for acceleration:

• m * a = F * (a - x) / (a - x)
• a = F / m

Thus, the magnitude of the acceleration of each particle when the separation becomes 2x is:

Final Thoughts

This result shows that the acceleration of the particles is directly proportional to the force applied at the midpoint and inversely proportional to their mass. This relationship is fundamental in understanding how forces translate into motion in systems involving tension and connected bodies.