Center of Mass

Every single physical system has associated itself with a certain point whose motion characterizes the motion of the whole system. When the system moves under an external force, then this point moves as if the entire mass of the system were concentrated at it, and also the external force was applied at it. This point is therefore called the ‘center of mass’ of the system. The motion of a system can be described in terms of the motion of its center of mass.   

While studying the dynamics of the movement of the system of particles as a whole, do not be bothered about the dynamics of an individual particle of the system. Rather focus only on the dynamics of a unique point corresponding to that system. The movement of this unique point is analogous to the movement of a single particle whose mass is equivalent to the sum of all independent particles of the system and the outcome of all the forces exerted on all the particles of the system by surrounding bodies (or) action of a field of force is exerted directly to that particle. This point is known as the center of mass of the system of particles. The idea of the center of mass (COM) is functional in analyzing the complicated movement of the system of objects, especially when two and more objects collide or an object explodes into fragments.

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What is Center of Mass?

A position that is defined relative to an object or a system of objects is known as the Centre of Mass of that object. In a system, the average position of all parts that are weighted according to their Masses is defined as the Centre of Mass. The Centre of Mass for the simple rigid objects having uniform density is located at the Centroid. It is possible that sometimes the Centre of Mass of an object does not lie over it. In the case of a uniform disc, the Centre of Mass would lie on its Centre, while the Centre of Mass in the case of a ring lies on the Centre but there is not any material. 

The Centre of Mass for complicated shapes can be defined as the unique position where the sum of the weighted position vectors of all the parts equals zero. 

What's a Rigid Body?

In everyday terms, we often come across things that can bend or stay stiff. These things can be soft, like a rubber ball, or hard, like a metal rod. When something can change its shape, we say it's deformable. But if it stays the same no matter what, we call it rigid.

Imagine an object made up of lots of tiny particles really close together. When this object changes its shape, the particles move around, and their distances from each other change. Now, a rigid body is like a special object where no matter what you do to it, the distances between all those tiny particles stay constant.

Think of it as the average position of all the parts of this object, taking into account how much each part weighs. For a simple rigid thing with an even density (like the same amount of stuff spread out evenly), the center of mass is right at the middle of it.

Importance of Centre of Mass

The Centre of Mass of a system is that one point where any uniform force is acted upon the object. It is important to find the Centre of Mass of objects as it makes it easy to solve the Mechanics’ problems in order to describe the Motion of complicated and oddly shaped objects. While doing calculations, we assume that all the Mass of an oddly-shaped object is concentrated in a tiny object that is located at the Centre of Mass, and this tiny object that we have assumed is known as the point Mass.

The formula for the Centre of Mass

The Vector Addition of the weighted position vectors that point towards the Centre of Mass of each object, gives the Centre of Mass of that system. The Centre of Mass is calculated separately for the components along each axis. 

For x-axis:

\[x_{com}=\frac{m_1x_1+m_2x_2+m_3x_3+...}{m_1+m_2+m_3+...}\]

For y-axis:

\[y_{com}=\frac{m_1y_1+m_2y_2+m_3y_3+...}{m_1+m_2+m_3+...}\]

\[x_{com}\] and \[y_{com}\] together will give the coordinates for the Centre of Mass of a system.

Center of mass formula for point objects: 

Calculating the Center of mass isn't a juggling act, but a systematic process. Here's how to find the Center of mass for different types of objects:

Discrete Mass Systems: For a system with a finite number of point masses (think planets orbiting a star), use the formula

\[x_{com}=\dfrac{\sum_{i=0}^{n}m_ix_i}{M}\]

where x_cm is the x-coordinate of the Center of mass, m_i are the individual masses, and x_i are their respective x-coordinates. 

Center of mass formula for any geometric shape: 

\[x_{com}=\frac{l}{2}\]

Continuous Mass Distribution: For objects like uniform rods or discs, where mass is continuously distributed, resort to integration. Express mass as a function of position and integrate over the object's extent to find the average position.

$x_{com} = \int{ \dfrac{x dm}{M}}$

$y_{com} = \int{ \dfrac{y dm}{M}}$

$z_{com} = \int{ \dfrac{z dm}{M}}$

Note: Symmetrical objects often have their Center of mass at the geometrical center.

Center of Mass of a Two-Particle System

Let us consider a system of two particles of masses m1 and m2 located at points A and B respectively.             

Let r1 and r2 be the position vectors of the particles relative to a fixed origin O. Then, the position vector rcm of the center of mass C of the system is defined by 

(m1+m2) rcm =m1 r1+m2 r2.

The product of the total mass of the system and the position vector of the center of mass is equal to the sum of the products of the masses of the two particles and their respective position vectors.  

Center of Gravity

The imaginary point through which on an object or a system, the force of Gravity is acted upon is known as the Centre of Gravity of that system. Usually, it is assumed while doing mechanical problems that the gravitational field is uniform which means that the Centre of Gravity and the Centre of Mass is at the same position. 

Properties of Center of Mass:

• Invariance: The Center of Mass remains stationary relative to the reference frame if the object undergoes only translational motion (no rotation).

• Additivity: The Center of Mass of a system is the vector sum of the individual Center of Masses of its constituent parts, weighted by their masses.

• Coincidence with centroid: For uniform-density objects, the Center of Mass coincides with the geometrical centroid.

• Motion under uniform force: If a uniform force acts on an object, it will cause a translational acceleration without any angular acceleration, as if the entire mass were concentrated at the Center of Mass.

Applications of Center of Mass:

• Solving translational motion problems: By treating the object as a point mass at the Center of Mass, we can simplify calculations involving linear momentum, kinetic energy, and collisions.

• Understanding rocket dynamics: The Center of Mass plays a crucial role in analyzing the motion and stability of rockets, as their mass distribution changes during fuel burn.

• Explaining planetary motion: The Center of Mass of the solar system lies slightly outside the Sun due to the presence of other planets, influencing their orbital dynamics.

• Analysing acrobatics and sports: The concept of Center of Mass helps explain the balance and maneuvers of athletes in various sports like gymnastics, diving, and pole vaulting.

Problem-Solving Techniques of Centre of Mass:

1. Direct calculation: For simple objects with known geometry and uniform density, the Center of Mass coordinates can be directly calculated using the formulas for geometric centroids.

2. Particle method: For complex objects or systems, the Center of Mass can be determined by dividing the object into smaller particles and calculating the weighted average of their positions.

3. Integral calculus: For continuous mass distributions, the Center of Mass can be found using integral calculus to sum the contributions of infinitesimal mass elements.

Solved Example

Example 1: m₁=2kg and m₂=5kg are two-point Masses located at y₁=10m and y₂=-5m respectively. Calculate the Centre of Mass.

We know that

 \[y_{com}=\frac{m_1y_1+m_2y_2+...}{m_1+m_2+...}\]

\[y_{com}=\frac{2\times 10+5\times (-5)}{2+5}\]

\[y_{com}=\frac{20-25}{7}\]=\[-\frac{5}{7}\]

Therefore, \[y_{com}=-\frac{5}{7}\]

Example 2: In the HCL molecule, the separation between the nuclei of the two atoms is 1.27 A. Find the location of the center of mass of the molecule. A chlorine atom is 35.5 times heavier than a hydrogen atom.   

Solution: The center of mass of the HCL molecule will be on the line joining H and CL atoms. Let the HCL molecule be along the X-axis, the H atom being at the origin (x=0). The center of mass relative to the H atom is given by 

\[x_{cm}=\frac{m_1x_1+m_2x_2}{m_1+m_2}\]

Where m1 and m2are the respective masses of H and CL atoms, and x1and x2 their distances relative to the H atom. 

Here, m2=35.5 m1, x1=0 and x2=1.27 A

\[x_{cm}=\frac{(m_1\times0)+(35.5m_1\times1.27A)}{m_1+35.5m_1}\]=\[\frac{35.5\times1.27A}{36.5}\]=1.235A

Example 3: Two bodies of masses 0.5 kg amnd 1 kg are lying in the X-Y plane at points (-1, 2) and (3,4) respectively. Locate the center of mass of the system.  

Solution : Here, we have 

m1=0.5 kg, m2=1 kg, x1= -1, y1=2, x2=3, y2=4.

By definition, the coordinates of the center of mass of the two bodies are given by: 

\[x_{cm}=\frac{m_1x_1+m_2x_2}{m_1+m_2}\]=\[\frac{(0.5)(-1)+(1)(4)}{(0.5)+(1)}\]=\[\frac{5}{3}\]

\[y_{cm}=\frac{m_1y_1+m_2y_2}{m_1+m_2}\]=\[\frac{(0.5)(2)+(1)(4)}{(0.5)+(1)}\]=\[\frac{10}{3}\]

The coordinates of C>M> are (5/3, 10/3).

Follow and practice more center of mass examples to get a better understanding.

Conclusion

The center of mass is a powerful concept that unlocks a deeper understanding of motion and equilibrium. By mastering this concept, you'll be well-equipped to tackle JEE Main Physics problems and gain valuable insight into the fascinating world of mechanics. Remember, the center of mass is not just a point on a page – it's the master of motion, guiding the dance of objects in our universe.