Question

From a uniform disk of radius R, a circular hole of radius R/2 is cut out. The centre of the hole is at R/2 from the centre of the original disc. Locate the centre of gravity of the resulting fiat body.

Answer 1

Let the initial mass entire disk = M

∴ Average mass/unit area = \(\frac {M}{(π R^2)}\)

mass of the unit cut out = \(\frac {M}{(π R^2)}\) π (R/2)²

∴ Mass of remaining planar object = M - \(\frac{M}{4}\) =\(\frac {3M}{4}\)

Consider region 1. it has a mass \(\frac{M}{4}\) and has a centre of mass (R/2,0)

Let the plane 2 has a centre of mass at (X_cm, Y_cm) and has a man of (3M/4)

For the entire disc of radius R, the CM lies at (R, O)

From the definition, R_cm = \(\frac {∑m_ir_i}{∑m_i}\)

∴ The new CM is at (7R/6,0).