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Replace Each Piece by a Point Mass at Its Own Centre of Mass

The centre of mass is the mass-weighted average position. For composite bodies, treat each piece as a point mass at its own centre of mass; for bodies with a missing part, treat the missing part as a negative mass.

NSEJS · Collisions and Centre of Mass · 9 linked questions · 100% coverage

Know this first

Position of the Centre of Mass is the central idea in this question pattern. Identify the condition in the stem before applying the method below.

Study deeper: NCERT Science · Collisions and Centre of Mass

Point masses and standard bodies

R_{cm} = \sum m r / \sum m

Equal masses M at (0, a), (0, 0) and (a, 0) give the centre of mass at (a/3, a/3). A 5 kg mass at (5, 4) and a 10 kg mass at (4, 3) give (65/15, 50/15) = (13/3, 10/3). Standard positions measured from the flat base: solid cone H/4 (5 cm for H = 20 cm), solid hemisphere 3R/8 (9 cm for R = 24 cm), hollow hemisphere R/2. The centre of mass of the pieces of a body lies closer to the heavier piece, so when a cricket bat is cut at its centre of mass, the piece with the shorter lever arm, the blade end, is the heavier one.

Composite and non-uniform bodies

Split a lamina into rectangles or squares and place each piece's mass at its geometric centre. An L-shaped lamina made of pieces of mass m at (a, 3a), 2m at (a, a) and 3m at (3a, a) has x_cm = (m a + 2m a + 9m a)/6m = 2a and y_cm = (3m a + 2m a + 3m a)/6m = 4a/3.

For a rod with density

ρ = ρ0 x²/L², x_cm = ∫ x ρ dx / ∫ ρ dx = (L⁴/4)/(L³/3) = 3L/4, shifted toward the denser end

Bodies with a part removed

Treat the hole as a negative mass at its own centre of mass. From a circular plate of radius 28 cm, a circle of radius 21 cm is removed from one edge. The removed area is (21/28)² = 9/16 of the whole, so 7/16 remains. The hole's centre is 28 - 21 = 7 cm from the plate's centre, so the remaining body's centre of mass lies on the far side: x_cm = -(9/16)(7)/(7/16) = -9 cm, that is 9 cm from the centre away from the hole.

A uniform sheet of mass 4m from which a mass m has been cut at a/2 above the original centre moves its centre of mass to -m (a/2) / 3

m = -a/6

OlympIQ exam lens

✓R_cm = Σ m r / Σ m; weights are masses, not areas, unless the density is uniform
✓Standard values from the base: cone H/4, solid hemisphere 3R/8, hollow hemisphere R/2
✓Composite body: place each piece's mass at its own centre of mass
✓Removed part: negative mass at its centre; the centre of mass moves away from the hole
✓Continuous bodies: x_cm = ∫ x dm / ∫ dm

OlympIQ trap

Do not average positions without weighting by mass. Three pieces of different mass need their masses as weights, and a removed piece enters with a minus sign, not as a zero.

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