Introduction to Center of Mass

Overview

This lecture is a quick and effective review of the "Center of Mass" chapter, summarizing the main concepts, formulas, and key points asked in exams.

The center of mass is the point where the entire system's mass can be considered concentrated at a single point (true in some cases, false in others).
Previously in physics, the whole body was considered as a particle; rotation and center of mass move towards real-life physics.
Rotation and center of mass are interconnected; studying both together is essential.

For discrete particles:
( X_{CM} = \frac{m_1x_1 + m_2x_2 + \dots}{m_1+m_2+\dots} )
Position vector:
( \vec{R}_{CM} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2 + \dots}{m_1+m_2+\dots} )
Continuous mass distribution:
( X_{CM} = \frac{\int x,dm}{\int dm} )
Remember formulas for center of mass of common bodies like rods, disks, circular rings, semi-rings, solid/hollow cones, etc.

Velocity:
( V_{CM} = \frac{m_1v_1 + m_2v_2 + \ldots}{m_1 + m_2 + \ldots} )
Acceleration:
( a_{CM} = \frac{m_1a_1 + m_2a_2 + \ldots}{m_1 + m_2 + \ldots} )
Total momentum of the system:
( P_{system} = (m_{total}) \cdot V_{CM} )
If net external force is zero, the system's momentum is conserved (initial = final).

In collisions:
Initial momentum = Final momentum (if external force is zero)
Coefficient of restitution (e):
( e = \frac{velocity, of, separation}{velocity, of, approach} )
Impulse (J):
( J = \Delta p = p_{final} - p_{initial} = \int F,dt )
Perfectly inelastic collision: both bodies stick together; maximum KE loss occurs.

In problems like platform+man, apply logic for changing/not changing the position of center of mass.
If there is no external force in x-direction, the center of mass will not move in the x-direction.
In oblique collisions, break velocity into line of impact and perpendicular components; apply head-on collision principle along LOI.
To find displacement of system's center of mass:
( M_1D_1 + M_2D_2 = 0 ) (if velocity of CM is initially 0)
Maximum compression in spring and mass system occurs when velocities of both masses become equal.

Center of Mass — the point where the entire system's mass can be considered concentrated.
Impulse — a large force applied in a very short time that changes momentum.
Coefficient of Restitution (e) — ratio of separation speed to approach speed, from 0 (inelastic) to 1 (elastic).
LOI (Line of Impact) — the normal line during collision on which normal force acts.
Lambda (λ) — mass per unit length.
Sigma (σ) — mass per unit area.
Rho (ρ) — mass per unit volume.