Centre of Mass for Continuous Mass Distribution
Determining the centre of mass for rigid bodies with continuous mass distribution using integration techniques.
What is Centre of Mass for Continuous Mass Distribution?
The average position of all the mass in a system or object. It is the point where the object would balance if suspended.
Key formula / rule: Position Vector of Centre of Mass
Key points
- Understand the concept of centre of mass for continuous bodies.
- Apply integration techniques to find the centre of mass.
- Derive the centre of mass for simple continuous objects (rods, discs, spheres).
- Differentiate between geometric center and centre of mass.
Common exam trap
Incorrectly setting up the integral limits.
Definitions
- Term
Centre of Mass (CM)
- Meaning
The average position of all the mass in a system or object. It is the point where the object would balance if suspended.
- Term
Infinitesimal Mass Element (dm)
- Meaning
A very small, theoretically infinitesimal portion of the total mass of a continuous object, used in integration to calculate properties like the centre of mass.
- Term
Linear Density (λ)
- Meaning
Mass per unit length of an object. λ = Mass/Length.
- Term
Surface Density (σ)
- Meaning
Mass per unit area of an object. σ = Mass/Area.
- Term
Volume Density (ρ)
- Meaning
Mass per unit volume of an object. ρ = Mass/Volume.
Learning objectives
Understand the concept of centre of mass for continuous bodies.
Apply integration techniques to find the centre of mass.
Derive the centre of mass for simple continuous objects (rods, discs, spheres).
Differentiate between geometric center and centre of mass.
Formulae
- Name
Position Vector of Centre of Mass
- Note
M is the total mass, r is the position vector of the infinitesimal mass element dm.
- Expression
Rcm = (1/M) ∫ r dm
- Name
Centre of Mass along X-axis
- Note
x is the x-coordinate of the infinitesimal mass element dm.
- Expression
Xcm = (1/M) ∫ x dm
- Name
Centre of Mass along Y-axis
- Note
y is the y-coordinate of the infinitesimal mass element dm.
- Expression
Ycm = (1/M) ∫ y dm
- Name
Centre of Mass along Z-axis
- Note
z is the z-coordinate of the infinitesimal mass element dm.
- Expression
Zcm = (1/M) ∫ z dm
- Name
Infinitesimal Mass Element (Volume)
- Note
ρ is the density, dV is the infinitesimal volume element. Used for 3D bodies.
- Expression
dm = ρ dV
- Name
Infinitesimal Mass Element (Area)
- Note
σ is the surface density, dA is the infinitesimal area element. Used for 2D bodies.
- Expression
dm = σ dA
- Name
Infinitesimal Mass Element (Length)
- Note
λ is the linear density, dL is the infinitesimal length element. Used for 1D bodies.
- Expression
dm = λ dL
Prerequisites
Concept of Centre of Mass for discrete bodies
Basic Calculus (Integration and Differentiation)
Understanding of Vectors
Density and its variations
Common mistakes
Incorrectly setting up the integral limits.
Failing to express dm correctly in terms of coordinates and density.
Confusing geometric center with center of mass for non-uniform density.
Errors in integration or differentiation.
Not considering the coordinate system properly.
Keywords
Centre of Mass
Continuous Distribution
Integration
Infinitesimal Mass Element
Density
Rigid Body Dynamics
Calculus
Practice preview
Which mathematical technique is primarily used to determine the centre of mass for a continuous mass distribution?…
easy
What is the position of the centre of mass of a uniform semicircular ring of radius R, with its centre at the origin?…
medium
A non-uniform rod of length L has a linear mass density given by λ(x) = ax, where 'a' is a constant and x is the distance from one end. What is the position of its centre of mass from the end where x=0?…
medium