Question

A homogeneous circular disc of radius r and weight W hangs in a vertical plane from a point O at its circumference. Find the period for small angles of swing in the plane of the disc.

• A. $2\pi \sqrt{\frac{r}{g}}$
• B. $2\pi \sqrt{\frac{3r}{2g}}$
• C. $\pi \sqrt{\frac{3r}{2g}}$
• D. $2\pi \sqrt{\frac{r}{3g}}$

Answer 1

The correct option is B $2\pi \sqrt{\frac{3r}{2g}}$

The period of oscillation of the compound pendulum

$T=2\pi \sqrt{\frac{L}{g}}$

where L is the equivalent length of the pendulum

$L=c+\frac{i_g^2}{c}$

c = distance of c.g. from the axis of rotation = r

Moment of inertia of the disc about an axis through its centre of gravity and perpendicular to the plane of the disc

$I_G=\frac{W}{g}{r}^2/2$

$\therefore$ Radius of gyration of the disc about its centroidal axis

$i_g^2=r^2/2$

$\therefore L=c+\frac{i_g^2}{c}=r+\frac{r^2/2}{r}=\frac{3}{2}r$

$\therefore T=2\pi \sqrt{\frac{\frac{3r}{2}}{g}}=2\pi \sqrt{\frac{3r}{2g}}$