A system consists of two small disks, of masses $ m $ and $ 2m $, attached to a rod of negligible mass of length $ 3l $ as shown above. The rod is free to turn about a vertical axis through point $ P $. The two disks rest on a rough horizontal surface; the coefficient of friction between the disks and the surface is $ \mu $. At time $ t = 0 $, the rod has an initial counterclockwise angular velocity $ \omega_0 $ about $ P $. The system is gradually brought to rest by friction. Develop expressions for the following quantities in terms of $ \mu $, $ m $, $ l $, $ g $, and $ \omega_0 $.

Question

A system consists of two small disks, of masses $ m $ and $ 2m $, attached to a rod of negligible mass of length $ 3l $ as shown above. The rod is free to turn about a vertical axis through point $ P $. The two disks rest on a rough horizontal surface; the coefficient of friction between the disks and the surface is $ \mu $. At time $ t = 0 $, the rod has an initial counterclockwise angular velocity $ \omega_0 $ about $ P $. The system is gradually brought to rest by friction. Develop expressions for the following quantities in terms of $ \mu $, $ m $, $ l $, $ g $, and $ \omega_0 $.(a) The initial angular momentum of the system about the axis through $ P $
, (b) The frictional torque acting on the system about the axis through $ P $
, (c) The time $ T $ at which the system will come to rest.
:

Jason · Accepted Answer

$\boxed{6ml^2 \omega_0}$
$\tau = 4\mu mgl$. A negative torque is also acceptable.
$\boxed{\frac{3l\omega_0}{2\mu g}}$

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