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Question

A solid cylinder of mass m and radius R is kept in equilibrium on a horizontal rough surface. Three unstretched springs of spring constants k, 2k, 3k are attached to cylinder as shown in the figure. Find period of small oscillations. (Given that surface is rough enough to prevent slipping of cylinder.)
1045880_26465ebd520643d98431b95c404bfa52.png


Solution

Let there is sufficient friction to support pure rolling
Let the cylinder rotates by an angle $$\theta$$ clockwise
Compression in the upper spring$$=(2R)\times(\theta)$$
Compression in the middle spring$$=(R+\dfrac{R}{2})\times(\theta)=\dfrac{3R\theta}{2}$$
Compression in the lower spring$$=(R)\times(\theta)$$
Torque applied by upper spring$$=F\times Leverarm=k\times2R\theta\times2R=4kR^2\theta$$
Torque applied by middle spring$$=F\times Leverarm=2k\times\dfrac{3}{2}R\theta\times\dfrac{3}{2}R=\dfrac{9}{2}kR^2\theta$$
Torque applied by lower spring$$=F\times Leverarm=3k\times R\theta\times R=3kR^2\theta$$
Net anticlockwise torque$$=4kR^2\theta+\dfrac{9}{2}kR^2\theta+3kR^2\theta=\dfrac{23}{2}kR^2\theta$$
Taking the torque about the lowest point of the cylinder$$=I\alpha=\dfrac{23}{2}kR^2\theta$$
$$\Rightarrow\left(\dfrac{mR^2}{2}+mR^2\right)\alpha=\dfrac{23}{2}kR^2\theta$$
$$\Rightarrow\left(\dfrac{3}{2}mR^2\right)\alpha=\dfrac{23}{2}kR^2\theta$$
$$\Rightarrow 3m\times\alpha=23k\theta$$
$$\Rightarrow \alpha=\dfrac{23k\theta}{3m}$$
$$\alpha=\omega^2\theta=\dfrac{23k\theta}{3m}\Rightarrow\omega=\sqrt{\dfrac{23k}{3m}}$$
$$\omega=\dfrac{2\pi}{T}=\sqrt{\dfrac{23k}{3m}}$$
$$T=2\pi\sqrt{\dfrac{3m}{23k}}$$

Physics

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