Question

You have a particle hold on a ceiling by a spring with elastic constant "k" and the length of the spring is not streched. The mass of the particle attached to the spring and the spring is massless and it is at rest. Consider the axis with the origin in the ceiling find the equation of the extension in function of the time.

Answer 1

This is a spring-mass problem with a change of origin.
If the unextended length of the spring is L the mass will move down a distance A to the static equilibrium position when the force up due to the extension of the spring is equal to the force down due to the weight of the mass.
The mass will then overshoot that position because it has kinetic energy and move a further distance A, stop and then move back until the spring is again unextended when the cycle will be repeated.
So this is a spring-mass simple harmonic motion problem with a change of origin.
You consider the shm about the equilibrium position with a displacement y.
Now y = -A when t=0 and the velocity at t=0 is also zero.
\[y=-A \cos(\omega t) \;\;\text{where} \;\;\omega = \sqrt{\dfrac{k}{m}}\]

From the ceiling the equilibrium position is x = L + A away.

The rest in on the attachment.
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