Springs+Cylinders mounted on an Axle-2!

Classical Mechanics Level 5

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 $(b)$ :-

Try part 1 also - Springs+Cylinders mounted on an Axle!

\frac{\Delta l}{R} \frac{d}{\sqrt{d^2 + 2R^2}} \sqrt{\frac{k}{m}}

\frac{\Delta l}{R} \frac{d}{\sqrt{d^2 + 2R^2}} \sqrt{\frac{2k}{m}}

\frac{\Delta l}{R} \frac{2d}{\sqrt{d^2 + R^2}} \sqrt{\frac{k}{m}}

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

None of These

1 solution

Nishant Rai
Jun 10, 2015

$\omega_1$ is angular velocity of disk.

$\omega_2$ is angular velocity of rod.

Applying Angular momentum conservation

$(m \frac{\omega_2 d}{2} \frac{d}{2})*2 = (\frac{mR^2}{2} \omega_1 )*2$

$\Rightarrow \omega_2 = 2\omega_1 \frac{R^2}{d^2}..........(1)$

Now apply $\text{Conservation of Energy}$ and evaluate the final answer using $(1)$

I think this is not a correct way to conserve angular momentum. You should consider the moment of inertia of the entire system about the center of the rod. Instead you are considering the disks as a point mass. According to me (m (w2) *d d/4 +m (w2) *R R) 2= (m R R (w1) /2)*2

Saiyam Bharara - 2 years, 2 months ago

Yeah,I also think so and then proceeding with further calculations "none of these" will be the best option

raj abhinav - 1 year, 3 months ago

Springs+Cylinders mounted on an Axle-2!

1 solution

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