Question

A uniform disc of radius $R$ is put over another uniform disc of radius $2R$ of same thickness and density. The peripheries of the two discs touch each other. The position of their center of mass is:

• A. at $\frac{R}{3}$ from the centre of the bigger disc towards the centre of the smaller disc
• B. at $\frac{R}{5}$ from the centre of the bigger disc towards the centre of the smaller disc
• C. at $\frac{2R}{5}$ from the centre of the bigger disc towards the centre of the smaller disc
• D. at $\frac{2R}{5}$ from the centre of the smaller disc towards the centre of the smaller disc

Answer 1

The correct option is B at $\frac{R}{5}$ from the centre of the bigger disc towards the centre of the smaller disc

Given,

Smaller disc radius, $=R$

Larger disc radius, $=2R$

Density of disc $=\rho$

Thickness of disc$=t$

Mass of larger disc and smaller disc, $m_L\&m_s$

Assume center of mass at distance x, from center of larger disc toward center of smaller disc.

From, formula of Center of Mass

$m_s\times R+m_L\times 0=(m_s+m_L)\times x$

$x=\frac{m_s\times R}{m_s+m_L}=\frac{\rho \left(\pi R^{2}t\right)\times R}{\rho \left(\pi R^{2}t\right)+\rho \left[\pi {\left(2R\right)}^{2}t\right]}=\frac{R}{5}$

center of mass at distance $\frac{R}{5}$, from center of larger disc toward center of smaller disc