Question

Consider a uniform circular disk of radius $R=5\,cm$ , pivoted smoothly at its centre of mass. As shown in the below figure, a spring of constant $K=5\,N/m$ is connected to it. If the mass of the disk is $m=2\,kg$ , then the angular frequency of torsional oscillations of the disc when it is given a small disturbance of $\omega\, rad/s$ . Write the value of $\left[\omega \right],$ where $\left[\right]$ is the greatest integer function.
(Take $\sqrt{5}=2.23$ )

Answer 1

If disc is twisted clockwise for a small angle $\theta $ , deformation in spring $x=R\theta $
Spring force $F_{s}=kx=k\left(\right.R\theta \left.\right)$
Restoring torque $\tau_{res}=-F_{s}R$
$\Rightarrow \left(\tau\right)_{res}=-\left(kR\right)^{2}\theta ....\left(i\right)$
Also from the laws of rotation, $\tau=I_{c}\alpha ....\left(ii\right)$
Here $I_{c}$ is the moment of inertia of the disc about the axis passing through its centre and perpendicular to the plane.
Hence, $I_{c}=\frac{m R^{2}}{2}$
From equations $\left(i\right)$ & $\left(ii\right)$ ,
$\alpha =\frac{kR^{2} \theta }{I_{c}}$
$\Rightarrow \alpha =-\frac{2 k}{m}\theta $
Comparing with the standard equation of angular SHM,
$\omega ^{2}\theta =\frac{- 2 k}{m}\theta $
$\therefore \omega =\sqrt{\frac{2 k}{m}}$
$\Rightarrow \omega =\sqrt{\frac{2 \times 5}{2}}$
$\therefore \omega =2.23\,rad/s$
Hence, $\left[\omega \right]=2$