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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
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Verified Answer
The correct answer is:
$2 \pi \sqrt{\frac{3 M}{8 K}}$
Let $x$ be the extension in the spring, Then, $x=2 R \theta$
$\therefore$ Torque acting on the cylinder about point of contact,
$T=F \times 2 R=-K x(2 R)=-4 R^2 K \cdot \theta$
$\therefore T=2 \pi \sqrt{\frac{I}{K e}}=2 \pi \sqrt{\frac{\frac{3}{2} M R^2}{4 R^2 K}}=2 \pi \sqrt{\frac{3 M}{8 K}}$
$\therefore$ Torque acting on the cylinder about point of contact,
$T=F \times 2 R=-K x(2 R)=-4 R^2 K \cdot \theta$
$\therefore T=2 \pi \sqrt{\frac{I}{K e}}=2 \pi \sqrt{\frac{\frac{3}{2} M R^2}{4 R^2 K}}=2 \pi \sqrt{\frac{3 M}{8 K}}$
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