Spinning rod dropping on a pivot

[mjolnir80]

Homework Statement

a long, thin rod of mass M and length L is standing stright up on a table. its lower end rotates on a frictionless pivot. a very slight push causes the rod to fall over. as it hits the table, what are (a) the angular velocity and (b) the speed of the tip of the rod?

Homework Equations

The Attempt at a Solution

could we set up a conservation of energy equation for this?
1/2I(for spinning of rod)$$\omega$$^2 + mgh = 1/2 I(for spinning of rod)$$\omega$$^2 + 1/2I(for dropping of rod)$$\omega$$^2

 

[LowlyPion]

I'd suggest that there is only rotational kinetic energy to consider.

And it gets fed by the PE of the center of mass doesn't it?

 

[nlightner]

Yes, we can use the conservation of energy equation to solve for the angular velocity and speed of the rod. We can also use the conservation of angular momentum equation, as the angular momentum of the rod will be conserved during the fall.

To solve for the angular velocity, we can use the conservation of energy equation you provided. We can set the initial energy (when the rod is standing straight up) equal to the final energy (when the rod is falling and rotating on the pivot). The initial energy will consist of the potential energy due to the height of the rod and the kinetic energy due to the spinning of the rod. The final energy will consist of the kinetic energy due to the rotation of the rod after it has fallen.

To solve for the speed of the tip of the rod, we can use the conservation of angular momentum equation. Since the angular momentum of the rod will be conserved during the fall, we can set the initial angular momentum (when the rod is standing straight up) equal to the final angular momentum (when the rod is falling and rotating on the pivot). The initial angular momentum will be zero, as the rod is not rotating initially. The final angular momentum will consist of the angular momentum due to the spinning of the rod and the angular momentum due to the rotation of the rod after it has fallen.

By setting up and solving these equations, we can determine the angular velocity and speed of the tip of the rod when it hits the table.