Motion of Centre of Mass for Physics Class 11

The centre of mass is basically the point where any object balances itself. All the mass of the object concentrates at that specific point. That helps in analysing forces and motion. It's like the sweet spot that you fix when wearing a backpack, so that it does not pull you backward or forward. Unlike the centre of gravity, which can change based on how strong gravity is in different places, the centre of mass solely depends on where the mass is located in the object. 

Centre of Mass Formulas

In the sixth Physics Chapter of Class 11, you will have to be clear about finding the centre of mass for two types of systems, as they have different formulas.  

For a discrete system, the centre of mass formula is x_cm = (Σm_i × x_i) / Σm_i. Here you sum up each mass times its position and divide by the total mass. 

For a continuous system, the centre of mass formula is x_cm = (∫x dm) / ∫dm. Here you are finding the average position of all the mass using integration.

Once you're clear with these two calculations, the next step is to move into figuring out how the centre of mass works when a real-world object moves. A continuous system like a hammer spins through the air when you throw it. It rotates wildly. Yet its path becomes quite predictable because you can easily look into the centre of mass, where all the mass of the hammer is concentrated at one point. That would have a parabolic path, just like a projectile. 

 

The motion of a centre of a mass is the movement of the balance point of an object, where all the mass of the object concentrates there. This balance point moves as if all the external forces are acting at that point, and that we need to calculate in physics Class 11.  

Effect of Internal vs External Forces on the Motion of the Centre of Mass

Whether a system comprises two particles or a complex rigid body, the COM moves as if all external forces act at this point. It follows Newton's laws, primarily the second and third. This concept is particularly useful in separating translational and rotational motion in rigid bodies. 

The only focus is on translational motion for centre of mass, as there are no internal forces at play. Because of Newton's Third Law, the forces here cancel out. Then we are left with external forces that influence the COM's acceleration.

Centre of Mass in Translational vs Rotational Motion 

The motion of a centre of mass follows the principle of conservation of momentum, in the linear way, as the movement is translational. 

It's important to remember that the motion of a centre of mass primarily involves the translational behaviour of a system. We do not have to consider the internal dynamics or rotational motion for now. Because for an object that is going through translational motion, there can be a simultaneous rotational motion to it as well.

But this rotational motion is governed by angular velocities and moment of inertia. You will come across these concepts later in the sixth chapter of Physics Class 11.  

For a system of $n$  particles with masses $m_1,m_2,\dots ,m_n$  at position vectors ${\vec{r}}_1,{\vec{r}}_2,\dots ,{\vec{r}}_n$ , the centre of mass' (COM's) position is:

${\vec{r}}_{\mathrm{C}\mathrm{O}\mathrm{M}}=\frac{\underset{i=1}{\overset{n}{\sum }} m_i{\vec{r}}_i}{\underset{i=1}{\overset{n}{\sum }} m_i}=\frac{m_1{\vec{r}}_1+m_2{\vec{r}}_2+\cdots +m_n{\vec{r}}_n}{M}$

where $M=m_1+m_2+\cdots +m_n$ is the total mass.

Velocity of Centre of Mass

The velocity of the centre of mass is obtained by differentiating. 

${\vec{v}}_{\mathrm{C}\mathrm{O}\mathrm{M}}=\frac{d{\vec{r}}_{\mathrm{C}\mathrm{O}\mathrm{M}}}{dt}=\frac{\underset{i=1}{\overset{n}{\sum }} m_i{\vec{v}}_i}{M}$

Acceleration of Centre of Mass

The acceleration of the COM is: ${\vec{a}}_{\mathrm{C}\mathrm{O}\mathrm{M}}=\frac{d{\vec{v}}_{\mathrm{C}\mathrm{O}\mathrm{M}}}{dt}=\frac{\underset{i=1}{\overset{n}{\sum }} {\vec{F}}_i^{\mathrm{e}\mathrm{x}\mathrm{t}}}{M}$

By Newton's second law, the net external force drives the COM's motion: ${\vec{F}}_{\mathrm{e}\mathrm{x}\mathrm{t}}=M{\vec{a}}_{\mathrm{C}\mathrm{O}\mathrm{M}}$

Internal forces cancel out due to Newton's third law, leaving only external forces to affect the COM.

That's how system's linear momentum is: $\vec{P}=\underset{i=1}{\overset{n}{\sum }} m_i{\vec{v}}_i=M{\vec{v}}_{\mathrm{C}\mathrm{O}\mathrm{M}}$

The motion of the centre of mass is closely linked to the conservation of momentum.

We know from the Laws of Motion chapter that momentum conservation describes an isolated system’s total momentum is constant when no external forces are acting on it. 

When there are no external forces acting ( ${\vec{F}}_{\text{ext}}=0$  ), the COM's acceleration is zero. Likewise, its velocity remains constant:

${\vec{v}}_{\mathrm{C}\mathrm{O}\mathrm{M}}=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$

This also implies the conservation of linear momentum ( $\vec{P}=M{\vec{v}}_{\text{COM}}$ ). M is the total mass. 

Example of How the Motion of a Centre of Mass is Related to Momentum Conservation

Two skaters push off from one another on an ice rink. Consider there is negligible friction.  Then, their individual velocities may change, showing that there is motion of the centre of mass.  

But the centre of mass of the system keeps moving at a constant velocity, and there are no forces until negligible friction takes over.  

That shows us conservation of momentum in action.

 

As you move to advanced levels, you have to learn that the motion of a centre of mass also varies by system. 

Isolated Systems: With ${\vec{F}}_{\text{ext}}=0$ , the COM's velocity is constant, as in collisions. 

Non-Isolated Systems with External Forces: Gravity or applied forces accelerate the COM. Generally, you can think of a falling body’s COM that follows a parabolic path. Here again you can go back to projectile motion. 

Rigid and Extended Bodies: The COM translates while the body rotates about it. That is in combined motion.

Example: A system of two particles with masses $m_1=2\mathrm{k}\mathrm{g}$  and $m_2=3\mathrm{k}\mathrm{g}$  at velocities ${\vec{v}}_1=4\hat{i}\mathrm{m}/\mathrm{s}$  and ${\vec{v}}_2=-2\hat{j}\mathrm{m}/\mathrm{s}$ . The COM velocity is:

${\vec{v}}_{\mathrm{C}\mathrm{O}\mathrm{M}}=\frac{m_1{\vec{v}}_1+m_2{\vec{v}}_2}{m_1+m_2}=\frac{\left(2\right)\left(4\hat{i}\right)+\left(3\right)(-2\hat{j})}{2+3}=\frac{8\hat{i}-6\hat{j}}{5}=1.6\hat{i}-1.2\hat{j}\mathrm{m}/\mathrm{s}$

In competitive exams, including JEE Main, just be thorough with the following. 

• Velocity Calculations: Use ${\vec{v}}_{\text{COM}}$ for multi-particle systems. The motion in isolated systems is translational, where linear momentum conservation rules apply.  
• Momentum Conservation: Analyse collisions or explosions, for which the principles of momentum conservation are related. 
• Trajectory Analysis: Determine the COM's path under external forces. Projectiles are important to remember here.
• Combined Motion: Separating translational and rotational dynamics, which you will learn as you advance in this chapter.

 

Here are all the Physics Class 11 notes. 

Units and Measurements Class 11 Notes	Mechanical Properties of Solids Class 11 Notes
Motion in a Straight Line Class 11 Notes	Mechanical Properties of Fluids Class 11 Notes
NCERT Class 11 Notes for Motion in a Plane	Thermal Properties of Matter Class 11 Notes
Laws of Motion Class 11 Notes	Thermodynamics Class 11 Notes
Work, Energy, and Power Class 11 Notes	Kinetic Theory of Gas Class 11 Notes
System of Particles and Rotational Motion Class 11 Notes	Oscillations Class 11 Notes
Gravitation Class 11 Notes	Waves Class 11 Notes

Get an overview of notes for Science in CBSE board.  

NCERT Class 11 Notes for PCM

NCERT Class 11 Physics Notes