Question

Consider a uniform circular disk of radius $R=5cm$ , pivoted smoothly at its centre of mass. As shown in the below figure, a spring of constant $K=5N/m$ is connected to it. If the mass of the disk is $m=2kg$ , 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

Answer: 2

If disc is twisted clockwise for a small angle $\theta$ , deformation in spring $x=R\theta$
Spring force $F_s=kx=k(R\theta )$
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 $\left(i\right)$ & $\left(ii\right)$ ,
$\alpha =\frac{k{R}^2\theta }{I_c}$
$\Rightarrow \alpha =-\frac{2k}{m}\theta$
Comparing with the standard equation of angular SHM,
${\omega }^2\theta =\frac{-2k}{m}\theta$
$\therefore \omega =\sqrt{\frac{2k}{m}}$
$\Rightarrow \omega =\sqrt{\frac{2\times 5}{2}}$
$\therefore \omega =2.23rad/s$
Hence, $\left[\omega \right]=2$