Centre of Mass

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The centre of mass (COM) is a basic concept in mechanics, describing a unique point in a system of particles or a rigid body where the entire mass can be considered to be act for the purpose of analysing motion.

Definition of Centre of Mass

The centre of mass is defined as the point where the total mass of a system or object is concentrated. Mathematically, it is the weighted average position of all the mass elements in a system. The motion of a system of particles can be described by considering the motion of its centre of mass.
For a system of n particles with masses m1​,m2​,…,mn​​located at positions r1,r2,…,rn​ the position of the centre of mass Rcm​ is given by: Rcm​  =  i = 1n​mi​ ri  / i = 1 mi​​
For a rigid body, the formula integrates over the continuous mass distribution:
Rcm = 1 / M r dm
Here, M is the total mass, r is the position vector, and dm represents an infinitesimal mass element.
Centre of Mass-mass
Mass

Physical Significance

1.Simplifies Analysis: The centre of mass reduces a complex system of particles into a single point for studying translational motion. This is especially useful when dealing with extended bodies or systems in motion.
2. Equilibrium and Stability: In static systems, the position of the centre of mass determines stability. If the centre of mass lies within the base of support, the system remains stable.
3. Rotational Motion: The centre of mass plays a key role in analysing rotational dynamics. When torque acts on a body, it often rotates around its centre of mass.
Centre of Mass-Position
position

Centre of Mass in Various Systems

1.For Two Particles: If two particles of masses m1​ and m2​ are separated by a distance d, the centre of mass lies closer to the heavier particle. Its position along the line joining the masses is:
xcm = m1.x1+  m2. x2​​ / m1 +m2
2. Uniform Rod: For a uniform rod, the mass is distributed evenly, and the centre of mass is at its geometric center.
3. Symmetrical Bodies: For objects with uniform density and symmetrical shapes (spheres, cylinders, cubes), the centre of mass coincides with their geometric centre.
4. Irregular Bodies: For irregular objects, the centre of mass can be found by dividing the object into small elements, calculating their individual moments, and summing them.

Motion of the Centre of Mass

The centre of mass obeys Newton’s laws of motion. For a system of particles, the external force Fext​ determines the acceleration of the centre of mass: Fext = Macm​
This implies that the motion of the centre of mass is influenced only by external forces, regardless of internal interactions within the system.

Applications

1.Projectile Motion: The trajectory of a projectile can be analysed using the centre of mass. Even if the object rotates or changes shape mid-air, its centre of mass follows a parabolic path.
2. Spacecraft Dynamics: In space missions, the centre of mass is critical for ensuring stability and control of spacecraft.
3.Engineering and Robotics: Designing stable structures and mobile robots often in calculating the centre of mass for balance and efficiency.
Centre of Mass-Space
Space

Experimental Determination

For simple objects, the centre of mass can be found by suspending the object from different points. The intersection of the vertical lines through the suspension points gives the centre of mass. For complex systems, computational methods are used.

Note

The centre of mass is a flexible and essential concept. It provides a simplified way to analyse motion, stability, and dynamics for systems of particles and rigid bodies. Its study not only enriches our mechanics but also has deep suggestion in various scientific and engineering fields.
The centre of mass is the point in a system of particles or a rigid body where the entire mass can be considered to be concentrated for the purpose of analyzing motion. It represents the weighted average position of all the mass elements in the system.
For a system of nnn particles with masses m1,m2,…,mn​ at positions r1,r2,…,rn​,…,rn​, the position of the centre of mass R⃗cm\vec{R}_{\text{cm}}Rcm​ is given by: Rcm​  =  i = 1n​mi​ ri  / i = 1 mi​​
Where ri​ is the position vector of the i-th particle.
No, the centre of mass does not always lie within the physical boundaries of the object. For instance, in a hollow ring or an irregular shape, the centre of mass may lie outside the object.
The centre of mass simplifies the study of motion by allowing the system of particles to be treated as if all its mass is concentrated at this single point. It is useful for analyzing translational motion, equilibrium, and rotational dynamics.
The motion of the centre of mass is governed by the net external force acting on the system. Its acceleration is given by:
acm = Fext / M​​
Where Fext​ is the total external force, and M is the total mass of the system.
For a uniform rod, where the mass is distributed evenly along its length, the centre of mass lies at its geometric centre, which is the midpoint of the rod.
The position of the centre of mass determines the stability of an object. An object is stable if its centre of mass lies within its base of support. If the centre of mass moves outside this base, the object becomes unstable and may topple.

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