A homogenous chain of mass density ρ and length L is kept vertical above a horizontal level. We let the chain free. What is the maximum reaction force of the horizontal level ?
Well, we can solve this problem easily by writing down Newton's II Law for the chain. That is F - ρLg = mυ' + m'υ. But if we name x the height of the chain x=x(t), then the mass m that goes into the momentum is m=ρx. Thus, F - ρLg = (ρx)x'' + ρυ^2. The acceleration x'' is equal to the gravitational acceleration (x''=-g) and thus the velocity υ is given by υ^2 = 2g(L - x). So plugging that into II Law, we get F = 3ρg( L - x).
So the maximum force is when x = 0, and is equal to F=3ρgL=3Mg. The reaction force is maximum just before the chain stops, and immediately after the reaction force balances the weight of the chain (F=Mg).
Comments
Post a Comment