An ideal spring hangs from the ceiling. A 1.25 kg mass is hung from the spring, stretching the spring a distance 𝑑=0.0895 m from its original length when it reaches equilibrium

Question

An ideal spring hangs from the ceiling. A 1.25 kg mass is hung from the spring, stretching the spring a distance 𝑑=0.0895 m from its original length when it reaches equilibrium. The mass is then lifted up a distance 𝐿=0.0275 m from the equilibrium position and released. What is the kinetic energy of the mass at the instant it passes back through the equilibrium position?

Answer 1

So Elena H.,

Solve this problem bit by bit.

1) Determine the value of the spring constant of the spring. Remember, F = k x ?

2) The rest position of the mass (hanging motionless on the spring) is your "zero" of displacement. But NOT the "zero" of potential energy of the spring, THAT is at the unstretched position of the spring!

3) You have lifted the mass by a certain distance. That entails gravitational potential energy acquired, doesn't it. How much? (The spring is irrelevant to that calculation).

4) Now for the harder conceptual part! What energies come into play as the mass falls to the rest position? Gravitational potential energy being released, yes. If you ALSO imagined (as I hope you did) that the spring was being stretched, and that that process was storing energy in the spring, you are most of the way there. For a stretched spring, integrating F with respect to distance to obtain work done (= energy stored) gives W = (1/2) k x^2 . So calculate how much energy that is, at two positions: where the mass was lifted to, and where the hanging rest position is. The difference between these two (spring) energies is how much (additional) work is stored in the spring as it descends as in the problem -- I say "additional" since there already was some stored at the release position. And the kinetic energy that the mass acquires as it "free" falls is the DIFFERENCE between the gravitational potential energy released and the work done on (== energy additionally stored in) the spring. Take the difference. There you are!

-- Cheers, --Mr. d.