Question

Let's assume two-point masses of mass m and 2m connected by a rigid massless rod. Initially, they both are at rest, then if we provide an impulse perpendicular to the rod to mass m let's say, the system will be in motion. The motion is restricted by a few rules: first is that according to rigid body constraint, the distance between both the mass should be constant for all times t. Now let's define a point called the center of mass such that the acceleration is zero of the point and we are doing it because it's a nice property. we can define the point like this acₘ = ma₁ + 2ma₂/m₁ + m₂. Now this definition allows acₘ to be zero due to newtons third laws because the force applied on the first mass is negative of the force applied on the second mass and their accelerations are inversely proportional to their masses so the numerator is zero and acₘ is zero and if you integrate twice you get that point, our center of mass. So the motion of this system is just a rotation around the center of mass. But there is a huge problem with this. It's that the denominator can be anything it can be m₁-m₂ or any other expression. So what does this mean this means that according to our derivation, there is an infinite family of points such that when there is no external force to it it doesn't move. can someone help? Is there a mistake in one of my steps?

Answer 1

The center of mass is a unique point for which the acceleration is zero, given no external forces. The error is in considering multiple possible denominators; the correct denominator for the center of mass formula is the combined mass of the system.

Step-by-step explanation:

The concept of the center of mass (CM) is fundamental in understanding the motion of a system of particles. When considering a two-point mass system with masses m and 2m connected by a rigid rod, and applying an impulse to the mass m, we can model the system's behavior using Newton's laws. In particular, Newton's third law will imply that the internal forces acting between the masses cancel out when calculating the acceleration of the center of mass. Therefore, there is only one such point, the center of mass, for which the acceleration is zero when no external forces are applied.

The formula for the acceleration of the center of mass acm should correctly be given by acm = (m * a1 + 2m * a2) / (m + 2m), where a1 and a2 are the accelerations of masses m and 2m, respectively. The denominator should represent the total mass of the system, which in this case is m + 2m (or 3m). If there are no external forces, the center of mass will not accelerate (acm = 0), and thus the system's motion can be described as a rotation about the center of mass.