## Center of Mass – Class 11 | Chapter – 7 | Physics Short Notes Series PDF for NEET & JEE

Center of Mass: The center of mass (COM) is a point that represents the average position of the mass of an object or system of objects. It is a useful concept in physics, as it can be used to analyze the motion of objects or systems. The center of mass has several important properties. It always lies on a line connecting two points of opposite symmetry in the object, and it moves as if all the mass of the object were concentrated at that point. This allows us to simplify the analysis of the motion of complex systems, by treating them as point masses located at their center of mass.

## Center of Mass

The position of the center of mass depends on the distribution of mass within the object or system.

For a system of particles, the position of the center of mass is given by:

COM = (m1r1 + m2r2 + … + mnrn) / (m1 + m2 + … + mn)

Where,

• m1, m2, …, mn are the masses of the particles
• r1, r2, …, rn are their respective positions

For a continuous object with a density function ρ(x, y, z), the position of the center of mass is given by:

COM = (1/M) ∫∫∫ (xρ, yρ, zρ) dV

Where,

• M is the total mass of the object, and
• the integral is taken over the volume V of the object.

## Properties of Center of Mass

The center of mass (COM) has several important properties that make it a useful concept in physics. Some of the key properties of the center of mass include:

• The center of mass always lies on a line connecting two points of opposite symmetry in the object. This means that if an object has a line of symmetry, the center of mass will lie on that line. If the object has more than one line of symmetry, the center of mass will lie at their intersection.
• The center of mass moves as if all the mass of the object were concentrated at that point. This means that if an external force is applied to an object, the center of mass will accelerate as if the entire mass of the object were located there. This makes it a useful concept in analyzing the motion of complex systems, by treating them as point masses located at their center of mass.
• The motion of an object can be analyzed in terms of the motion of its center of mass. This is because the motion of the center of mass is related to the motion of the object as a whole. For example, if an object is rotating about a fixed axis, the motion of the center of mass is related to the rotational motion of the object.
• If an object is supported at its center of mass, it will be in a stable equilibrium. This is because any small disturbance to the object will cause the center of mass to move slightly, but the object will still be supported at that point. If an object is not supported at its center of mass, it will be in an unstable equilibrium and will tend to fall over.
• The position of the center of mass depends on the distribution of mass within the object. For example, if an object has more mass on one side than the other, the center of mass will be closer to the heavier side. If an object has a uniform density, the center of mass will be located at the geometric center of the object.

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