Motion of Centre of Mass — Revision Notes

⚡ 30-Second Revision

Position of CM: — $\vec{R}_{CM} = \frac{\sum m_i \vec{r}_i}{M}$
Velocity of CM: — $\vec{V}_{CM} = \frac{\sum m_i \vec{v}_i}{M} = \frac{\vec{P}_{sys}}{M}$
Acceleration of CM: — $\vec{A}_{CM} = \frac{\sum m_i \vec{a}_i}{M}$
Newton's 2nd Law for CM: — $\vec{F}_{ext} = M \vec{A}_{CM}$
Internal Forces: — Do NOT affect CM motion ($\sum \vec{F}_{int} = 0$)
External Forces: — ONLY affect CM motion.
Conservation of Momentum: — If $\vec{F}_{ext} = 0$, then $\vec{V}_{CM} = \text{constant}$ (and $\vec{P}_{sys} = \text{constant}$).

2-Minute Revision

The motion of the center of mass (CM) is a simplified way to describe the overall translational movement of a system of particles or an extended body. The CM's position is the mass-weighted average of all particle positions.

Its velocity, $\vec{V}_{CM}$, is the total momentum of the system divided by its total mass, $\vec{V}_{CM} = \vec{P}_{sys}/M$. Crucially, the acceleration of the CM, $\vec{A}_{CM}$, is determined solely by the net external force acting on the system, following $\vec{F}_{ext} = M \vec{A}_{CM}$.

Internal forces, which are forces between particles within the system, always cancel out and have no effect on the CM's motion. This means if there are no external forces, the CM's velocity remains constant (conservation of linear momentum).

This principle is vital for analyzing explosions (CM remains on original path), collisions (CM velocity is constant if isolated), and relative motion problems (e.g., man on a boat, where CM remains stationary horizontally).

Remember to treat all quantities as vectors and correctly identify internal vs. external forces.

5-Minute Revision

To master the motion of the center of mass for NEET, focus on these core ideas. The center of mass (CM) is a conceptual point that simplifies the analysis of complex systems. Its position is given by $\vec{R}_{CM} = \frac{\sum m_i \vec{r}_i}{M}$.

Differentiating this, we get the velocity of the CM: $\vec{V}_{CM} = \frac{\sum m_i \vec{v}_i}{M}$. This is equivalent to the total linear momentum of the system divided by its total mass, $\vec{P}_{sys}/M$.

Further differentiation yields the acceleration of the CM: $\vec{A}_{CM} = \frac{\sum m_i \vec{a}_i}{M}$.

The most critical concept is Newton's Second Law for a system of particles: $\vec{F}_{ext} = M \vec{A}_{CM}$. This states that only the net external force acting on the system can cause its center of mass to accelerate.

Internal forces, such as those from an explosion or interactions between colliding particles, always sum to zero vectorially and thus do not affect the CM's motion. For example, if a bomb explodes in mid-air, its CM continues along the original parabolic trajectory because gravity is the only external force.

If the net external force is zero, then $\vec{A}_{CM} = 0$, which implies $\vec{V}_{CM}$ is constant. This is the principle of conservation of linear momentum for the system, a powerful tool for problems involving collisions, explosions, or a person walking on a boat in still water.

Always ensure you are using vector addition and correctly identifying external forces when solving problems.

Prelims Revision Notes

1
Definition: — Center of Mass (CM) is the point where the entire mass of a system is considered to be concentrated for translational motion analysis.
2
Position Vector of CM: — For discrete particles, $\vec{R}_{CM} = \frac{m_1\vec{r}_1 + m_2\vec{r}_2 + \ldots + m_n\vec{r}_n}{m_1 + m_2 + \ldots + m_n} = \frac{\sum m_i \vec{r}_i}{M_{total}}$.
3
Velocity of CM: — $\vec{V}_{CM} = \frac{m_1\vec{v}_1 + m_2\vec{v}_2 + \ldots + m_n\vec{v}_n}{M_{total}} = \frac{\sum m_i \vec{v}_i}{M_{total}}$. This is also $\vec{V}_{CM} = \frac{\vec{P}_{system}}{M_{total}}$.
4
Acceleration of CM: — $\vec{A}_{CM} = \frac{m_1\vec{a}_1 + m_2\vec{a}_2 + \ldots + m_n\vec{a}_n}{M_{total}} = \frac{\sum m_i \vec{a}_i}{M_{total}}$.
5
Newton's Second Law for a System: — The net external force acting on a system is equal to the total mass of the system multiplied by the acceleration of its center of mass: $\vec{F}_{ext} = M_{total} \vec{A}_{CM}$.
6
Internal vs. External Forces: — Internal forces (forces between particles within the system) do NOT affect the motion of the center of mass. Their vector sum is zero. Only external forces (forces from outside the system) can change the velocity or acceleration of the CM.
7
Conservation of Linear Momentum: — If the net external force acting on a system is zero ($\vec{F}_{ext} = 0$), then the acceleration of the CM is zero ($\vec{A}_{CM} = 0$). This means the velocity of the CM ($\vec{V}_{CM}$) remains constant. Consequently, the total linear momentum of the system ($\vec{P}_{system} = M_{total} \vec{V}_{CM}$) is conserved.
8
Applications:

* Explosions: CM continues its original path if $\vec{F}_{ext} = 0$ (e.g., bomb exploding in air, CM follows parabola). * Collisions: CM velocity is constant before, during, and after collision if $\vec{F}_{ext} = 0$. * Man on Boat: If a man walks on a boat in still water, the horizontal CM of the man-boat system remains stationary (or moves with constant velocity if initially moving). Use $m_1\Delta x_1 + m_2\Delta x_2 = 0$ for displacement problems.

1
CM Location: — CM can be outside the physical body (e.g., ring, hollow sphere).

Vyyuha Quick Recall

CM's Rule: Can't Move Externally, Internally No Effect.

Can't Move Externally: Center of Mass motion is only affected by External forces.
Internally No Effect: Internal forces have No Effect on the CM's overall motion.