Spring Centered Collar on Smooth Rod


A 20 pound collar is constrained to slide on a smooth rod and held in position by three springs each attached to the collar, with spring constants of 10, 10 and 30 pounds per foot. The springs are unstressed when the collar is at position s=0. If the collar is displaced to s=0.5 feet and released from rest, determine the speed when s=0.


The three springs combine to act like a single spring with spring constant 50 pounds per foot. The energy stored in a compressed spring is 1/2*k*x2 where k is the spring constant and x is the displacement. At displacement s=0.5 feet, the potential energy is 25*0.52 = 6.25 foot pounds. At s=0 this energy will be entirely converted to kinetic energy so 1/2*m*v2=6.25. The mass m is 20/32.2 slugs or 0.62 slugs so v2=12.5/0.62 = 20.16 ft2/s2. The speed then is 4.5 f/s.

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