Springs+Cylinders mounted on an Axle!

Classical Mechanics Level 4

There are two homogeneous, solid disks of radius $R$ and mass $m$ mounted by two parallel, horizontal axes at the ends of a horizontal rod of negligible mass. The distance between these axes is d and the disks can freely rotate around them. The rod itself, with the disks mounted on it, can also freely rotate around a horizontal axis in it midpoint. (See the figures. All the three axes are perpendicular to the rod.)

On the rim of each disk there is a small pin, and between them there is a spring of spring constant $k$ which is initially compressed by $\Delta l$ . (The spring is in contact with the pins until it extends to its un-stretched position and then falls down.)

Find the angular velocity of the disks after we burn the thread that hold the spring in it compressed position, provided that its initial position corresponds to figure $(a)$ :-

Try part 2 also - Springs+Cylinders mounted on an Axle-2!

\frac{2\Delta l}{R} \sqrt{\frac{k}{m}}

\frac{\Delta l}{R} \sqrt{\frac{2k}{3m}}

\frac{\Delta l}{2R} \sqrt{\frac{k}{m}}

None of These

\frac{\Delta l}{R} \sqrt{\frac{k}{m}}

2 solutions

Andrew Song
Sep 10, 2015

Elastic energy = rotational kinetic energy of both discs

Nishant Rai
Jun 10, 2015

Due to angular momentum conservation there is no need of rotation of rod.

$\text{Total energy of spring = Rotational kinetic energy of two disks}$

$\dfrac 1 2 k{\Delta l}^2 = [\dfrac 1 2 (\frac{mR^2}{2}) \omega^2 ]*2$

Springs+Cylinders mounted on an Axle!

2 solutions

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