Position of Centre of Mass – Class 11

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It serves as the foundation for more complex topics in rotational dynamics and mechanics. Center of mass is the precise location within a body or system of particles where all of the system’s mass is thought to be concentrated. Since this point acts as though all external forces are acting directly on it, knowing where the center of mass is will be essential to forecasting how items will move.

Definition of  Position of Centre of Mass

When evaluating translational motion, the center of mass (COM) of a system is defined as the point at which the system’s mass can be conceived of as concentrated.
When examining a system’s overall motion, this is the point at which the system’s complete mass can be said to act. Mathematically, the following formula can be used to determine the location of the center of mass in a system of particles:
                                        m1​r1​ + m2​ r2​ +⋯ + mn​ rn​​
                           Rcom  =   __________________
                                        m1 + m2  + ….  +  mn
where m1,m 2,…,mn​ are the masses of individual particles in the system and r1, r2 ,…, rn ​ are their respective position vectors.
In a continuous mass distribution, the formula is replaced by an integral:
                                  RCOM = rdm​ / dm
where r⃗ represents the position vector of each infinitesimal mass element dm.

Properties of Centre of Mass

1.System Behavior: Motion of a single particle at the center of mass in a system of particles can be used to explain the motion of the entire system, and the force acting on this point equals the total external force acting on the system.
2. External Forces: The center of mass’s motion is only influenced by external forces. Newton’s third law states that internal forces, such as those between system particles, cancel out and have no effect on the center of mass’s location or velocity
3. Motion: If the system is isolated and not subject to any outside influences, the center of mass’s velocity stays constant. The law of conservation of momentum is being used here.
4. Symmetry: The center of mass of symmetrically shaped objects is located along the axis of symmetry. For instance, the center of mass is located at the geometric center of an evenly dense disk or sphere.

Examples of Centre of Mass

1. Two-Particle System: Consider two masses, m1 and m2, placed at positions r⃗1​ and r⃗2 The position of the centre of mass is given by:
RCOM ​=  ​m1​r1​ + m2​r2​​  / m1​+ m2
If  m1 = m2​, the centre of mass is exactly midway between the two masses.
2. Rigid Body: The center of mass of a uniform rod of length L and mass M is found at L / 2, or the rod’s halfway. The homogeneous mass distribution along the rod’s length is the cause of this
3. Irregular Bodies: Finding the center of mass in a body with non-uniform mass distribution or irregular shape requires more complicated calculations, frequently using calculus.

Centre of Mass in Daily Life

  • Balancing Objects: To maintain stability, we essentially aim to align the center of mass of an object with a point of support when we balance it. To help with balance, a tightrope walker, for example, lowers their center of mass by carrying a pole.
  • Sports: Center of mass is a notion used by athletes, such gymnasts and high jumpers, to maximise movement efficiency. During a performance, they may regulate their balance and trajectory by changing their body posture.
  • Vehicle Design: When constructing vehicles, engineers use the concept of the center of mass to assure stability and reduce the chance of tipping. For example, during high-speed turns, the center of mass of racing automobiles is kept low to improve control and balance.
Position of Centre of Mass -athletes
Note:-
Fundamental ideas like the center of mass make it easier to analyse complex particle and rigid body systems. It makes it simpler to understand and forecast how a system will behave in response to outside stimuli since it enables us to treat a system as though all of its mass were concentrated at one location.
When examining the motion of a rigid body or a system of particles, the center of mass is the location where all of the mass can be concentrated. When thinking about the system’s motion, this is the moment at which external forces are thought to be at work.

The position of the centre of mass is determined using the formula:

                                        m1​r1​ + m2​ r2​ +⋯ + mn​ rn​​
                           Rcom  =   __________________
                                         m1 + m2  + …. +  mn

where m1, m2,…,mn are the masses of individual particles, and r⃗1,r⃗2,…,r⃗n are their respective positions.

Key properties include:
  • Internal forces have no effect on the position or motion of the center of mass; it moves as if all exterior forces are acting upon it.
   • The center of mass of symmetric objects with uniform density       is located along the axis of symmetry
Center of mass of a system maintains its current velocity in the absence of external forces, allowing it to continue moving in that direction. Law of conservation of momentum is the cause of this behavior.
The center of mass of a uniform rod is located at its halfway. This results from the mass’s even distribution along its length.
It is possible for the center of mass to be located outside the body. For example, the center of mass of a ring or hollow item is found at its geometric center, which is outside of the object’s substance.
By treating the system as though all of its mass is concentrated at one location, the center of mass makes the understanding of complicated systems simpler. As a result, it is simpler to analyse the motion and forecast how outside influences may affect the system.

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