A solid cylinder is attached to a horizontal massless spring as...

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

Options

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

Correct answer

C. $2 \pi \sqrt{\frac{3 M}{8 K}}$

Step-by-step solution

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}}

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