Question

A circular hole of radius $1cm$ is cut off from a disc of radius $6cm$ . The centre of the hole is $3cm$ from the centre of the disc. Then the distance of the centre of mass of the remaining disc from the centre of the disc is

• A. $\frac{3}{35}cm$
• B. $\frac{1}{35}cm$
• C. $\frac{3}{10}cm$
• D. none of these

Answer 1

Answer: (A)

For the calculation of the position of centre of mass, cut off mass is taken as negative. The mass of disc is

${\text{m}}_1=\pi {\text{r}}_1^2\sigma$ $=\pi {\text{6}}^2\sigma =\text{36}\pi \sigma$
where, $\sigma$ = surface mass density. The mass of cut out portion is
${\left(\text{m}\right)}_{\text{2}}=\pi {\left(\text{1}\right)}^2\sigma =\pi \sigma$
${\text{x}}_{\text{cm}}=\frac{{\text{m}}_1{\text{x}}_{\text{1}}-{\text{m}}_2{\text{x}}_2}{m_1-m_2}$
Taking origin at the centre of disc,
$x_1=0,x_2=3cm$
${\text{x}}_{\text{cm}}=\frac{\text{36}\pi \sigma \times 0-\pi \sigma \times 3}{36\pi \sigma -\pi \sigma }$
$=\frac{-3\pi \sigma }{35\pi \sigma }=-\frac{3}{35}\text{cm}$