UY1: Pizza with a hole

Question: A circular pizza of radius R has a circular piece of radius R/4 removed from one side, as shown in the figure below. Where is the centre of mass of the pizza with the hole in it?

Let the centre of mass of the pizza with the hole in it be x.

From the definition of centre of mass,

$$x_{CM} = \frac{1}{M} \sum \, mx$$

Since the centre of mass of a WHOLE pizza is at the centre, xCM is 0.

$$0 = \frac{1}{M} \left\{ \sigma \left[ \pi \left( \frac{R}{4} \right)^{2} \left( – \frac{3}{4} R \right) \right] + \sigma x \left[ \pi \left( R \right)^{2} – \pi \left( \frac{R}{4} \right)^{2} \right] \right\}$$

Basically, the above equation is saying that the whole pizza is made up of two portions: the cut out piece and the left over piece. The centre of mass of the cut out piece is $-\left( \frac{3}{4} \right) R$ while the centre of mass of the left over piece is x.

Re-arranging and solving,

$$x = \left( \frac{1}{20} \right) R$$

Back To UY1: Sample questions for centre of mass (Set 1)