Angular Velocity of u Shape - Solving for Angular Velocity

[Kelschul]

Angular Velocity?

A thin uniform rod is rotating at an angular velocity of 9.37 rad/s about an axis that is perpendicular to the rod at its center. As the figure indicates, the rod is hinged at two places, one-quarter of the length from each end. Without the aid of external torques, the rod suddenly assumes a "u" shape, with the arms of the "u" parallel to the rotation axis. What is the angular velocity of the rotating "u"?

Before: The system is a single rod of mass M and length L rotating about an axis through its center.
After: The system consists of three parts; a rod of mass M/2 and length L/2 rotating about an axis through its center and two masses M/4 rotating at a distance L/4 from the axis. (Treat these masses as particles.)



I know I should use the conservation of momentum, moment of inerta for the rod, Net torque= Inertia x angular acceleration but I have no idea where to start!

Help me exam tomorrow!

 

[Galileo]

Indeed, conservation of momentum is the way to go.
The momentum before is $I_1\omega_1$ and after it's $I_2\omega_2$.
So ask yourself what's given, what can you calculate and what is the unknown?

 

[m2003]

I would advise you to first draw a free body diagram of the system before and after the change in shape. This will help you visualize the forces acting on each part of the system and make it easier to apply the equations of motion.

Next, you can use the conservation of angular momentum to solve for the final angular velocity. Since there are no external torques acting on the system, the initial angular momentum should be equal to the final angular momentum. You can also use the moment of inertia for each part of the system to calculate the total moment of inertia.

Finally, you can use the equation torque = moment of inertia x angular acceleration to calculate the angular acceleration of the system. With this, you can solve for the final angular velocity.

It is important to carefully consider the direction of the angular velocity and angular acceleration in your calculations, as they may be opposite in direction for different parts of the system.

I understand that you have an exam tomorrow, but it is important to take the time to understand the concepts and equations before attempting to solve the problem. Good luck!