A thin rod of length l stands vertically on a table. The rod begins to fall, but

Question

A thin rod of length l stands vertically on a table. The rod begins to fall, but its lower end does not slide. (a) Determine the angular velocity of the rod as a function of the angle f it makes with the tabletop. (b) What is the speed of the tip of the rod just before it strikes the table?
A thin rod of length l stands vertically on a table. The rod begins to fall, but its lower end does not slide. (a) Determine the angular velocity of the rod as a function of the angle f it makes with the tabletop. (b) What is the speed of the tip of the rod just before it strikes the table?

Explanation / Answer

There are two forces on the rod first one is gravitational force,W = mg at the center of the length(L) force exerted by the table on the end of the rod, which does not slip.


From the free body diagram Sum of the torques on the rod = (moment of inertia)(rod's angular acceleration)

The torque with respect to the stationary end is    t= (mg)[(L/2)cos(?)]
but    t = I a
   t= [(1/3)mL^2]a

From the above two equations (mg)[(L/2)cos(?)] = [(1/3)mL^2]a    The the angular velocity of the rod  
     a = (3/2)(g/L) cos(?)
We know that     v = at        =  [ (3/2)(g/L) cos(?)]   t the speed of the tip of the rod is      v = [ (3/2)(g/L) cos(?)] t

   

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