Question

The pulley shown in figure has a MOI I about its axis and mass m.Find the time period of vertical oscillations of its COM. The spring has a spring constant k and the string does not slip over the pulley

• A. $2\pi \sqrt{\frac{I/r^2+m}{4k}}$
• B. $1\pi \sqrt{\frac{I/r^2+m}{4k}}$
• C. $6\pi \sqrt{\frac{I/r^2+m}{4k}}$
• D. None of these

Answer 1

The correct option is A $2\pi \sqrt{\frac{I/r^2+m}{4k}}$

$2T=mg$ or $2ky=mg$ or $y=\frac{Mg}{2k}$
Writing energy
$U=\frac{1}{2}I{\omega }^2+\frac{1}{2}m{v}^2-mgx+\frac{1}{2}k{[\frac{mg}{2k}+2x]}^2$
$=\frac{1}{2}[\frac{I}{r^2}+m]v^2+\frac{m^2{g}^2}{8k}+2k{x}^2$
Put $\frac{dU}{dt}=0$ as the system is conservative,giving
$0=[\frac{I}{r^2}+m]\frac{vdv}{dt}+4kxv$
or $\frac{dv}{dt}=a=-\frac{4kx}{\left(\frac{I}{r^2}+m\right)}$
$\therefore$ $T=2\pi \sqrt{\frac{I/r^2+m}{4k}}$