The two bodies of mass m1 and respectively are tied to the...

The two bodies of mass $m_{1}$ and respectively are tied to the ends of a massless string, which passes over a light and frietionless pulley. The masses are initially at rest and then released. Then acceleration of the centre of mass of the system is (a) $\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right)^{2} g$ (b) $\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right)^{2}$ (c) $g$ (d) zero

The two bodies of mass $m_{1}$ and respectively are tied to the ends of a massless string, which passes over a light and frietionless pulley. The masses are initially at rest and then released. Then acceleration of the centre of mass of the system is
(a) $\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right)^{2} g$
(b) $\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right)^{2}$
(c) $g$
(d) zero

Key Concepts

Internal Forces Cancellation

When analyzing the motion of a system as a whole, internal forces such as tension in a string cancel out because they occur in equal and opposite pairs. This cancellation means that only external forces need to be considered when applying Newton’s second law to determine the acceleration of the center of mass.

Constraint Relations in Connected Bodies

In systems where bodies are connected by elements such as strings or rods, the motions of the bodies are interdependent. The constraint (like inextensibility of a string) imposes relationships between individual accelerations, often resulting in one body’s acceleration being the negative of the other’s. This interdependence is crucial for correctly expressing the individual motions in terms of a single variable.

Centre of Mass Acceleration

This concept involves determining the acceleration of a system’s center of mass by taking the weighted average of the individual accelerations of all the masses, typically obtained by summing the products of each mass and its acceleration and then dividing by the total mass. It highlights that while individual parts may accelerate differently, the overall motion of the center of mass is dictated by the net external force acting on the system.

Newton’s Second Law for Systems

Newton’s second law, when applied to a system of particles, states that the sum of the external forces acting on the system equals the total mass multiplied by the acceleration of the center of mass. This principle allows one to connect the forces acting on separate parts with the collective motion of the entire system.

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