Centre of Mass — Formula Sheet

(m1 + m2) r_cm = m1 r1 + m2 r2

r_cm = (m1 r1 + m2 r2) / (m1 + m2)

If r_cm = 0, then m1 r1 + m2 r2 = 0

r_cm = (1/M) Σ(mi ri)

x_cm = (1/M) Σ(mi xi), y_cm = (1/M) Σ(mi yi), z_cm = (1/M) Σ(mi zi)

dm = (m/l) dx

x_cm = (∫0 to l x (m/l dx)) / (∫ dm) = (1/(2l))(l^2 - 0)

x_cm = l/2

x_cm = (2/3)L

If one end of the rod is at the origin and the rod lies along the x-axis, the centre of mass is at x = l / 2.