One end of a thin rod is attached to a pivot, about which it can rotate without friction. Air resistance is absent. The rod has a length of 0.59 m and is uniform. It is hanging vertically straight downward. The end of the rod nearest the floor is given a linear speed v0, so that the rod begins to rotate upward about the pivot. What must be the value of v0, such that the rod comes to a momentary halt in a straight-up orientation, exactly opposite to its initial orientation?

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1. The initial kinetic energy of the bar, which is
(1/2) I wo^2 = (1/2)*M (L^2/3)*(vo/L)^2
= (1/6)*M vo^2

Must equal the increase in potential energy in the upside-down position, which is
M g L
since the center of mass raises by an amount L.

I is the moment of inertia of the rod about the pivot at the top.

M cancels out and you are left with
(1/6)vo^2 = g*L

Solve for vo

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2. a uniform rod of strenght b stands vertically uoright on a rough floor and tge tips over what is the rod's angular velocity when it hits the floor

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