A thin uniform rod is rotating at an angular velocity of 6.7 rad/s about an axis that is perpendicular to the rod at its center. As the drawing 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"?
To find the angular velocity of the rotating "u" shape, we can make use of the principle of conservation of angular momentum.
The angular momentum of the system before the rod assumes the "u" shape is given by the formula:
L_initial = I_initial * ω_initial
where L_initial is the initial angular momentum, I_initial is the initial moment of inertia, and ω_initial is the initial angular velocity.
Since the rod is rotating about an axis perpendicular to the rod's length and passing through the center, the moment of inertia of the rod is given by:
I_initial = (1/12) * m * L^2
where m is the mass of the rod and L is the length of the rod.
Given that the rod is rotating at an angular velocity of 6.7 rad/s, we can calculate the initial angular momentum.
Now, when the rod assumes the "u" shape, the moment of inertia changes. The moment of inertia of the "u" shape can be approximated by adding the moment of inertia of two thin rods with lengths equal to one-quarter of the original rod length.
The moment of inertia of each thin rod is given by:
I_u = (1/12) * m * (L/4)^2
And since there are two thin rods, the total moment of inertia of the "u" shape is:
I_total = 2 * I_u
Now, we can calculate the final angular momentum of the "u" shape using the formula:
L_final = I_total * ω_final
Since the system is isolated and there are no external torques acting on it, the angular momentum is conserved. Therefore, we can equate the initial and final angular momenta:
L_initial = L_final
Replacing the values with the given information, we have:
I_initial * ω_initial = I_total * ω_final
Solving for ω_final, we find:
ω_final = (I_initial * ω_initial) / I_total
Substituting the expressions for I_initial and I_total, we get:
ω_final = ((1/12) * m * L^2 * ω_initial) / (2 * (1/12) * m * (L/4)^2)
Simplifying the expression further:
ω_final = (4 * ω_initial) / 2
ω_final = 2 * ω_initial
Therefore, the angular velocity of the rotating "u" shape is twice the initial angular velocity, which means:
ω_final = 2 * 6.7 = 13.4 rad/s