Centre of Mass for a System of n Particles
Generalizing the calculation of the centre of mass for a system comprising multiple point masses.
What is Centre of Mass for a System of n Particles?
A unique point in a system of particles or a body where the entire mass of the system is considered to be concentrated for the purpose of analyzing its translational motion.
Key formula / rule: Position Vector of Centre of Mass
Key points
- Define the centre of mass for a system of n discrete particles.
- Calculate the position vector of the centre of mass for a given system of particles using the appropriate formula.
- Determine the individual coordinates (XCM, YCM, ZCM) of the centre of mass in a 2D or 3D system.
- Understand the physical significance and implications of the centre of mass in analyzing the motion of a system.
Common exam trap
Forgetting to divide by the total mass (Mtotal) in the denominator.
Definitions
- Term
Centre of Mass (CM)
- Meaning
A unique point in a system of particles or a body where the entire mass of the system is considered to be concentrated for the purpose of analyzing its translational motion.
- Term
Position Vector (r)
- Meaning
A vector that describes the position of a point in space relative to a chosen origin.
- Term
System of Particles
- Meaning
A collection of individual particles whose collective motion and interactions are being studied as a single entity.
Learning objectives
Define the centre of mass for a system of n discrete particles.
Calculate the position vector of the centre of mass for a given system of particles using the appropriate formula.
Determine the individual coordinates (XCM, YCM, ZCM) of the centre of mass in a 2D or 3D system.
Understand the physical significance and implications of the centre of mass in analyzing the motion of a system.
Relate the motion of the centre of mass to the net external forces acting on the system.
Formulae
- Name
Position Vector of Centre of Mass
- Note
rᵢ are position vectors of individual particles, Mtotal is the total mass of the system.
- Expression
RCM = (m₁r₁ + m₂r₂ + ... + mₙrₙ) / (m₁ + m₂ + ... + mₙ) = (Σ mᵢrᵢ) / Mtotal
- Name
X-coordinate of Centre of Mass
- Note
xᵢ are the x-coordinates of the particles.
- Expression
XCM = (m₁x₁ + m₂x₂ + ... + mₙxₙ) / (m₁ + m₂ + ... + mₙ) = (Σ mᵢxᵢ) / Mtotal
- Name
Y-coordinate of Centre of Mass
- Note
yᵢ are the y-coordinates of the particles.
- Expression
YCM = (m₁y₁ + m₂y₂ + ... + mₙyₙ) / (m₁ + m₂ + ... + mₙ) = (Σ mᵢyᵢ) / Mtotal
- Name
Z-coordinate of Centre of Mass
- Note
zᵢ are the z-coordinates of the particles.
- Expression
ZCM = (m₁z₁ + m₂z₂ + ... + mₙzₙ) / (m₁ + m₂ + ... + mₙ) = (Σ mᵢzᵢ) / Mtotal
- Name
Total Mass of the System
- Note
Sum of the individual masses of all particles in the system.
- Expression
Mtotal = m₁ + m₂ + ... + mₙ = Σ mᵢ
Prerequisites
Basic vector algebra (addition, subtraction, components).
Understanding of position vectors and coordinate systems.
Concept of mass and total mass.
Familiarity with summation notation (Σ notation).
Common mistakes
Forgetting to divide by the total mass (Mtotal) in the denominator.
Incorrectly performing scalar addition instead of vector addition for position vectors (rᵢ).
Confusing the centre of mass with the centre of gravity; they are identical only in a uniform gravitational field.
Assuming the centre of mass must always be located within the physical boundaries of the object or system.
Errors in sign conventions when dealing with coordinates in a multi-dimensional system.
Keywords
Centre of Mass
CM
system of particles
position vector
total mass
weighted average
translational motion
barycenter
momentum
NEET Physics
Practice preview
For a system of particles, where can its centre of mass (CM) be located?…
easy
Four particles, each of mass 'm', are placed at the four vertices of a square of side 'a'. What are the coordinates of the centre of mass of this system if one vertex is at the origin (0,0) and the sides are along the x …
medium
Two particles of masses 5 kg and 10 kg are moving with velocities v1 = (2i + 3j) m/s and v2 = (-i + 4j) m/s respectively. What is the velocity of the centre of mass of this two-particle system?…
hard