A pencil placed vertically on a table falls down. What will be the linear velocity of the middle of the pencil at the end of the fall if the pencil is 15 cm long?
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The amount of kinetic energy in the pencil just before it touches the ground is equal to the gravitational potential energy it lost while falling. That quantity is easy to compute, being simply m*g*h, where h=d/2 (the center of mass of the pencil started at height d/2 and ends at height 0). If we assume the tip of the pencil hasn't moved (this becomes a much more complex problem if there is friction between the tip and the table, etc) then all of that energy is now rotational kinetic energy, which is equal to 1/2*I*ω. I is the moment of inertia (1/12*m*d^2) and ω is the angular velocity. You know d, the mass terms cancel out, and once you solve for angular velocity you can determine linear velocity of the end.
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