Let us consider the system that is similar to the system from Problem HW10 #3. T

Question

Let us consider the system that is similar to the system from Problem HW10 #3. Two particles with masses M that are attached to a rod with mass 2M and can move along the rod (this motion occurs probably with some friction), as shown in Figure and below. The rod with these two particles can freely rotate in a horizontal plane around a vertical axis passing through point O. The axis of rotation is parallel to z-axis. There is no friction between the rod and the axis. The length of the rod is 4L, and the particles are initially located at the distance of L from the rotation axis. Please, note here that the system is not a rigid body. At time t = 0, the rod with two masses in their initial positions [See Figure a)] is rotating around a vertical axis with the initial angular velocity w0. At later time, the two particles slowly move from its initial positions to the end of the rod as shown in Figure which shows the final position of the particles. What is the final angular velocity w f = ?

Explanation / Answer

angular momentum conservation:

I initial=16mL^2/12+2mL^2=40mL^2/12=10mL^2/3

I final = 16mL^2/12+8mL^2=28mL^2/3

I*omega=constant

==> omega final=omega0*10/28=0.35 omega0

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