Centre of Mass of a Composite Body

The method

Any composite body — L-shape, T-shape, H-shape, or any irregular shape made of standard pieces — can be handled by:

1. Break the body into standard shapes (rectangles, triangles, circles, rods, etc.)
2. Find the mass and CM of each part
3. Apply the discrete system formula:

$x_{cm} = \frac{m_1 x_1 + m_2 x_2 + \cdots}{m_1 + m_2 + \cdots}, \quad y_{cm} = \frac{m_1 y_1 + m_2 y_2 + \cdots}{m_1 + m_2 + \cdots}$

For uniform bodies, mass is proportional to area (2D) or volume (3D), so mass ratios can be replaced by area or volume ratios.

Example 1: L-shaped plate

A uniform L-shaped plate formed by two rectangles:

• Rectangle 1: $4 \times 1$ cm, lying along the $x$-axis from $(0,0)$ to $(4,1)$
• Rectangle 2: $1 \times 3$ cm, standing vertically from $(0,0)$ to $(1,3)$ (but the $1\times1$ overlap is counted in rectangle 1)

Let surface density $\sigma = 1$ (arbitrary units).

Rectangle A: $4 \times 1$, mass $= 4$, CM at $(2, 0.5)$

Rectangle B: $1 \times 2$ (the part above $y=1$), mass $= 2$, CM at $(0.5, 2)$

Total mass $= 6$

$x_{cm} = \frac{4(2) + 2(0.5)}{6} = \frac{8+1}{6} = \frac{9}{6} = 1.5 \text{ cm}$

$y_{cm} = \frac{4(0.5) + 2(2)}{6} = \frac{2+4}{6} = 1 \text{ cm}$

CM at $(1.5, 1)$ cm — outside the material, in the empty corner of the L.

Example 2: T-shaped plate

A T-shape: horizontal bar $6 \times 1$ cm centered at the top, vertical stem $1 \times 3$ cm below.

Bar: $6 \times 1$, mass $= 6$, CM at $(3, 3.5)$ (if stem goes from $y=0$ to $y=3$, bar from $y=3$ to $y=4$)

Stem: $1 \times 3$, mass $= 3$, CM at $(3, 1.5)$

$y_{cm} = \frac{6(3.5) + 3(1.5)}{9} = \frac{21 + 4.5}{9} = \frac{25.5}{9} \approx 2.83 \text{ cm}$

By symmetry: $x_{cm} = 3$ cm.

Example 3: Disc with rectangular cutout

A uniform disc of radius $R$ with a small rectangle of mass $m_2$ removed from it. Original disc mass $= M$.

After removal, remaining mass $= M - m_2$.

If the rectangle's CM was at position $x_2$:

$x_{cm,remaining} = \frac{Mx_{disc} - m_2 x_2}{M - m_2} = \frac{M \cdot 0 - m_2 x_2}{M - m_2} = \frac{-m_2 x_2}{M - m_2}$

(assuming disc CM at origin)

This is the negative mass method — covered separately.

Key insight

The CM formula is linear — the CM of a composite is just the mass-weighted average of the CMs of the parts. No integration needed once you know the CM of each standard shape.