Finding the Centre of Mass

1. Choose a coordinate system (i.e. an x and y axis) and an origin.

2. Break the object up into simply-shaped pieces (rectangles, circles, spheres, etc.). The centre of mass of each piece is at the geometric centre of each piece.

3. Label each piece.

4. Determine the coordinates of the geometric centre (the CM) of each piece.

5. Determine the mass of each piece. The mass of each piece is directly proportional to the area of the piece (in 2D) or the volume (3D).

6. Record the data in a table. Calculate mixi, miyi , and mizi for each piece as necessary; mi is the mass of each piece and (xi, yi, zi) is the location of the centre of mass of the piece in your coordinate system.

7. Holes can be treated as objects with negative mass.

EXAMPLE: The diagram below shows a 'P'-shaped sign attached to a wall by a hinge and a wire. The sign is constructed of uniform material and has a mass m = 100.0 kg. The centre of the 'P' is a square 1m × 1m hole. Find the centre of mass of the sign.

The 'P' has an area of 10 m². Since it is uniform, each 1m² must have a mass of 10 kg.

X_cm = (Σm_ix_i)/M_total = 130/100 = 1.30 m

Y_cm = (Σm_ix_i)/M_total = 300/100 = 3.00 m

Alternate solution using negative mass

Note that the hole having an area of 1 m² has a mass of -10 kg. We 'fill' in the hole and subtract it later.

X_cm = (Σm_ix_i)/M_total = 130/100 = 1.30 m

Y_cm = (Σm_ix_i)/M_total = 300/100 = 3.00 m

Exactly as before.

Questions? mike.coombes@kwantlen.ca