Question

If m and 2m are two masses separated by a distance of 1 m are attracted by gravitational force. Then the acceleration of their centre of mass is?

Hint:

Internal forces can never change the velocity or acceleration of the center of mass of a system. Only an external force can do that.

Answer 1

Step 1: Understanding the Concept:
The dynamics of a system of particles are governed by Newton's Second Law applied to the center of mass. The acceleration of the center of mass (\(a_{cm}\)) depends \textit{only} on the net external force acting on the entire system.
Step 2: Key Formula or Approach:
\[ F_{net, external} = M_{total} \times a_{cm} \]
If the net external force is zero, then \(a_{cm} = 0\), regardless of internal interactions.
Step 3: Detailed Explanation:
Consider the two masses \(m\) and \(2m\) as forming a single closed system.
The only force mentioned is the mutual gravitational attraction between them.
Mass \(m\) exerts a gravitational pull on mass \(2m\). By Newton's Third Law, mass \(2m\) exerts an equal and opposite gravitational pull on mass \(m\).
These gravitational forces are purely internal to the two-mass system. They are an action-reaction pair that sum to zero when considering the system as a whole.
Assuming there are no other outside forces acting on the masses (like gravity from a planet, friction, etc., as none are stated), the net external force on the system is exactly zero.
\[ \sum F_{ext} = 0 \]
According to the equation of motion for the center of mass:
\[ \sum F_{ext} = (m + 2m) \cdot a_{cm} \]
\[ 0 = 3m \cdot a_{cm} \]
\[ a_{cm} = 0 \]
Even though the individual masses accelerate towards each other, their center of mass remains perfectly stationary (or continues in uniform motion if it was already moving).
Step 4: Final Answer:
The acceleration of their centre of mass is zero.