Question

A system consists of a disc and square with equal surface mass density as shown in figure. The centre of mass is at the point of contact.The relation between $L$ and $R$ is $L=(2\pi {)}^{n}R$.Then, find $n$.
(Answer up to two decimal places)

Answer 1

Suppose $\sigma$ is the uniform mass density.
Mass of the disc, $m_1=\sigma \pi R^2$
and mass of the square, $m_2=\sigma L^2$

Taking the origin at the point of contact, position of COM is given by,
$x_{COM}=\frac{m_1{x}_1+m_2{x}_2}{m_1+m_2}$

Here, $x_1=-R,x_2=L/2$
$x_{COM}=0$ [COM is at origin]
$\therefore 0=\frac{\sigma \pi R^2\times (-R)+\sigma L^2\times L/2}{\sigma (\pi R^2+L^2)}=\frac{-\pi \times 2R^3+L^3}{\pi R^2+L^2}$
$\Rightarrow L^3=2\pi R^3$
$\Rightarrow L=(2\pi {)}^{1/3}R$

Hence, $n=\frac{1}{3}=0.33$