An object of mass m and speed v0 strikes a rigid uniform rod of length l and mas

Question

An object of mass m and speed v0 strikes a rigid uniform rod of length l and mass mr that is hanging by a frictionless pivot from the ceiling. Immediately after striking the rod, the object continues forward but its speed decreases to v0 /2. The moment of inertia of the rod about its center of mass is Icm= (1/12)mr l 2. Gravity acts with acceleration g downward. a) For what value of v0 will the rod just touch the ceiling on its first swing? You may express your answer in terms of g , mr , m , and l . b) For what ratio mr/ m will the collision be elastic?

Explanation / Answer

conservation of momentum m*vo=m*(vo/2)+mr*v v=m*vo/2*mr then apply conservation of energy between lowest and top point 0.5*m*v^2+0.5*I*w^2=m*g*l/2 I= moment of inertia about hinge=m*l^2/12+m*l^2/4=m*l^2/3 w=v/l solving this v=sqrt(3gl/4) so vo=(2mr/m)*sqrt(3gl/4) for elstic collision vo/2=[(m-mr)/(m+mr)]*vo mr/m=1/3

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