Question

A solid cylinder is attached to a horizontal massless spring as shown in figure. If the cylinder rolls without slipping, the time period of oscillation of the cylinder is?

• A. $2\pi \sqrt{\frac{x}{g}}$
• B. $2\pi \sqrt{\frac{2M}{3K}}$
• C. $2\pi \sqrt{\frac{3M}{8K}}$
• D. $2\pi \sqrt{\frac{3M}{2K}}$

Answer 1

The correct option is C $2\pi \sqrt{\frac{3M}{8K}}$

Let x be the extension in the spring,
Then, $x=2R\theta$
$\therefore$ Torque acting on the cylinder about point of contact,
$T=F\times 2R=-Kx\left(2R\right)=-4R^{2}K\cdot \theta$
$\therefore T=2\pi \sqrt{\frac{I}{Ke}}=2\pi \sqrt{\frac{\frac{3}{2}M{R}^2}{4R^{2}K}}=2\pi \sqrt{\frac{3M}{8K}}$.