First slide

Methods to locate centre of mass

Question

Find the position of centre of mass of a uniform disc of radius R from which a hole of radius r is cut out. The centre of the hole is at a distance $\frac{R}{2}$ from the centre of the disc.

Difficult

A

$\frac{{Rr}^{2}}{2 (R^{2} - r^{2})} towards right of O$
B

$\frac{{Rr}^{2}}{2 (R^{2} - r^{2})} towards left of O$
C

$\frac{2 {Rr}^{2}}{(R^{2} + r^{2})} towards right of O$
D

$\frac{2 {Rr}^{2}}{(R^{2} + r^{2})} towards left of O$

Solution

For a disc, mass is proportional to surface area. So
Mass of cut portion: $m_{1} = k {πr}^{2}$ ,
Mass of remaining portion: $m_{2} = k π (R^{2} - r^{2})$
Apply $m_{1} x_{1} = m_{2} x_{2}$ where $x_{1} = \frac{R}{2}$ , and solve to get
$x_{2} = \frac{{Rr}^{2}}{2 (R^{2} - r^{2})}$

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