Spring Centered Collar on Smooth Rod

## Question:

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.
## Answer:

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*x^{2} where k is the spring constant and
x is the displacement. At displacement s=0.5 feet, the potential
energy is 25*0.5^{2} = 6.25 foot pounds. At s=0 this
energy will be entirely converted to kinetic energy so 1/2*m*v^{2}=6.25. The mass m is 20/32.2
slugs or 0.62 slugs so v^{2}=12.5/0.62 =
20.16 ft^{2}/s^{2}. The speed then is
4.5 f/s.
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