Centre of Mass of Continuous Bodies
For a continuous mass distribution, the centre of mass is defined by integrating position against the mass element and dividing by total mass, with symmetry used only when the density and geometry justify it.
Why this shows up in the exam
Finding centroids of rods, plates, cones, and solids · Replacing extended weight by an equivalent point in uniform gravity · Using symmetry to locate balance points
Learn the idea
A continuous body's centre of mass is the density-weighted average over all of its material. A continuous body's centre of mass is the density-weighted average over all of its material. Start from a clear axis, origin, body, and reference frame; the geometry and constraints then decide which rotational law is safe to use.
🧠 Memory hook: Integrate position weighted by how much mass is there.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- r_cm = (1/M) integral r dm — General centre-of-mass definition for a continuous body.
- dm = lambda dl = sigma dA = rho dV — Mass element chosen for line, area, or volume density.
How to approach it
- 1Exploit valid symmetry before integrating
- 2Write the correct mass element and limits
- 3Compute the first moment and verify dimensions and location
Common slip-ups that cost marks
- •Using geometric centre for a nonuniform body
- •Choosing an inconsistent density element
- •Forgetting to divide the first moment by total mass
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
Masses 1 kg and 3 kg lie at x = 0 and x = 4 m. Find the x-coordinate of their centre of mass.
More from Motion of System of Particles and Rigid Body
Conservation of momentum and angular momentum
The total linear and angular momentum of a system remains constant in the absence of external forces or torques, including during collisions and rotational motion.
Moment of inertia and radius of gyration
Moment of inertia quantifies how mass is distributed with respect to an axis of rotation, and the radius of gyration is a measure related to this distribution.
Torque and rotational equilibrium
Torque is the rotational analogue of force, causing angular acceleration, and equilibrium occurs when the net torque on a body is zero.
Center of mass: definition and calculation
The center of mass is the point representing the mean position of the mass in a system, and can be calculated for discrete particles or continuous bodies.
Rotational kinematics and dynamics
Rotational kinematics describes the motion of rotating bodies, while dynamics relates torque, angular acceleration, and rotational kinetic energy.
Centre of Mass of Discrete Particles
For discrete particles, the centre-of-mass position is the vector sum of each mass times its position divided by total mass; this point governs translation even when the particles move relative to one another.