Composite Bodies, Holes, and Cavities
The centre of mass of a composite body follows the ordinary mass-weighted formula when added pieces have positive mass and removed pieces are represented by negative mass of the same density.
Why this shows up in the exam
Off-centre holes in discs and spheres · Laminas assembled from simple shapes · Locating balance after machining a cavity
Learn the idea
A removed piece can be treated as negative mass when computing the centre of mass of what remains. A removed piece can be treated as negative mass when computing the centre of mass of what remains. 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: A hole is a negative copy of the missing material.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- r_cm = (sum s_i M_i r_i)/(sum s_i M_i) — Composite-body formula with s_i = +1 for material and -1 for a hole.
- M_remaining = M_full - M_removed — Remaining mass when a uniform piece is cut away.
How to approach it
- 1Choose one reference origin
- 2List full and removed pieces with signed masses
- 3Check that the centre shifts away from the removed material
Common slip-ups that cost marks
- •Giving the hole positive mass
- •Using unequal densities without adjusting masses
- •Measuring component centroids from different origins
🌟 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
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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.