Two identical masses \(m=0.25 \mathrm{~kg}\) are suspended from a vertical rod by two rigid bars of length \(L=20 \mathrm{~cm}\) and negligible mass (Fig. 7.17, ). When the system rotates around the vertical axis with constant angular velocity \(\omega\), there exists a configuration in which the angle \(\theta\) that the two rods form with respect to the vertical is constant in time and as a function of \(\omega\).

1. Assuming that \(\theta\) is constant during the rotation, determine the relationship between the angular velocity \(\omega\) and the angle \(\theta\), neglecting all frictional forces.

2. Determine the minimum angular velocity \(\omega_{0}\) for which \(\theta>0\).

Now suppose that the two masses \(m\) are subjected to a viscous force proportional to their velocity, \(\mathbf{F}=-\beta \mathbf{v}\), being \(\beta=3.0 \mathrm{~kg} / \mathrm{s}\), and that the system is kept in rotation by a motor with angular velocity \(\omega_{1}=14 \mathrm{rad} / \mathrm{s}\). Under these conditions, assuming the angle \(\theta\) to be constant during the rotation, calculate:

1. the work performed by the engine in each revolution to overcome the viscous friction;

2. the axial component \(L_{z}\) of the angular momentum of the system with respect to the suspension point of the bar.

3. Suddenly the engine stops and the system begins to slow down due to friction. Calculate the amount of time that it is necessary to \(L_{z}\) to decreased by \(90 \%\) with respect to the value calculated in the previous point.

Fig. 7.17 

Transcribed Image Text:

L m 3 L m