To determine the acceleration of the center of mass of a system consisting of two particles connected by a light inextensible string over a smooth pulley, we can analyze the forces acting on each particle and apply Newton's laws of motion. Let's break this down step by step.
Understanding the System
We have two particles with masses in the ratio of n:1. Let's denote the mass of the heavier particle as m1 = n and the lighter particle as m2 = 1. When the system is released, the heavier mass will accelerate downward while the lighter mass will accelerate upward.
Forces Acting on Each Particle
- For m1 (mass n): The force acting downward is its weight, which is F1 = n * g, where g is the acceleration due to gravity.
- For m2 (mass 1): The force acting upward is its weight, which is F2 = 1 * g = g.
Net Force and Acceleration
When the system is released, the net force acting on the system can be calculated as follows:
Net Force (F_net) = Weight of m1 - Weight of m2
Substituting the values, we get:
F_net = n * g - g = (n - 1) * g
Calculating the Total Mass
The total mass of the system is the sum of the masses of both particles:
Total Mass (M) = m1 + m2 = n + 1
Using Newton's Second Law
According to Newton's second law, the acceleration (a) of the system can be expressed as:
a = F_net / M
Substituting the expressions we derived:
a = [(n - 1) * g] / (n + 1)
Acceleration of the Center of Mass
The acceleration of the center of mass (COM) of the system is the same as the acceleration of the entire system since both particles are connected. Therefore, the acceleration of the center of mass is:
a_COM = [(n - 1) * g] / (n + 1)
Final Thoughts
This result shows how the acceleration of the center of mass depends on the mass ratio of the two particles. If n is greater than 1, the heavier mass will accelerate downwards, causing the center of mass to move in that direction. Conversely, if n is less than 1, the lighter mass will dominate the motion. This principle illustrates the fundamental concepts of dynamics and the behavior of connected systems in physics.
