Center of mass Knowledge-Based Study

Study notes

1. The center of mass (CM) of a system is the unique point where the entire mass of the system can be considered concentrated for the purpose of describing its translational motion.

   For a system of $n$ discrete particles with masses $m_1, m_2, \dots, m_n$ located at positions $\vec{r}_1, \vec{r}_2, \dots, \vec{r}_n$, the position vector of the center of mass, $\vec{r}_{\text{cm}}$, is defined as:

   $$\vec{r}_{\text{cm}} = \frac{\sum_{i=1}^{n} m_i \vec{r}_i}{M_{\text{total}}}$$

   where $M_{\text{total}} = \sum_{i=1}^{n} m_i$. Along a single axis, this simplifies to:

   $$x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2 + \dots + m_n x_n}{m_1 + m_2 + \dots + m_n}$$

   The center of mass acts as a mass-weighted average position of the constituent particles.

   Differentiating the position of the center of mass with respect to time yields the velocity of the center of mass, $\vec{v}_{\text{cm}}$:

   $$\vec{v}_{\text{cm}} = \frac{d\vec{r}_{\text{cm}}}{dt} = \frac{\sum_{i=1}^{n} m_i \vec{v}_i}{M_{\text{total}}}$$

   Notice that the numerator represents the vector sum of the linear momenta of all individual particles in the system, defined as the total linear momentum $\vec{P}_{\text{sys}}$:

   $$\vec{P}_{\text{sys}} = \sum_{i=1}^{n} \vec{p}_i = \sum_{i=1}^{n} m_i \vec{v}_i$$

   Therefore, the total linear momentum of any system is directly related to the motion of its center of mass by:

   $$\vec{P}_{\text{sys}} = M_{\text{total}} \vec{v}_{\text{cm}}$$

   Differentiating the total momentum with respect to time leads to Newton's second law for a system of particles:

   $$\frac{d\vec{P}_{\text{sys}}}{dt} = M_{\text{total}} \vec{a}_{\text{cm}} = \sum \vec{F}_{\text{net}}$$

   By Newton's third law, all internal interaction forces between particles within the system occur in equal and opposite action-reaction pairs, summing vectorially to zero:

   $$\sum \vec{F}_{\text{int}} = \vec{0}$$

   Consequently, only net external forces can accelerate the center of mass:

   $$\sum \vec{F}_{\text{ext}} = M_{\text{total}} \vec{a}_{\text{cm}}$$

   Internal forces—such as collision contact forces, chemical reactions, or internal explosions—can never alter the acceleration or velocity of the center of mass.

   When an isolated system experiences zero net external force ($\sum \vec{F}_{\text{ext}} = \vec{0}$), its total linear momentum is strictly conserved:

   $$\vec{P}_{\text{sys}} = \text{constant} \implies \vec{v}_{\text{cm}} = \text{constant}$$

   This principle has significant physical implications:

   • Collisions: Whether a collision is elastic, inelastic, or completely inelastic, $\vec{v}_{\text{cm}}$ remains identical before, during, and after the impact provided no external impulses act.
   • Explosions: If a projectile in mid-air explodes due to internal forces, the center of mass of all fragments continues along the exact original parabolic trajectory until external forces such as ground impacts intervene.
   • Zero-Momentum Reference Frame: In the center-of-mass frame of reference, the total momentum of the system is identically zero at all times ($\vec{P}'_{\text{sys}} = \vec{0}$), making it an especially powerful frame for analyzing complex multi-body collisions.