Question

A particle of mass m is suspended from the ceiling by a spring of spring constant k and relaxed length l_ {o}, whose mass is negligible. The particle is released at rest with the spring relaxed. Taking the Oz axis directed vertically downward, with the origin on the roof, calculate the z position of the particle as a function of time

Answer 1

Well lets start with the basics. I hope you would agree with the statemet "the mass pulls the spring down/extends it/moves in the y downward or '-y direction.'"

Now we know what direction it will move in but not the answer to the question 'How Fast?'
Well Hooke's law tells us that it will be a constant downward speed. So a graph of 'distance the mass has been pulled down vs how long its been since we let the mass go' will be a straight line and if you incorporate the info from the first paragraph, you would realize the mass will move downward and it will do so at a constant rate. This is where the spring constant 'k' comes in. Make up linear (no exponents) equation with this info to get a function of force over time.

What about gravity?
The force on the spring is equal to the force of gravity acting on the mass. This force is m*g where m=mass and g= the gravity constant on earth at approx. the surface. F=m*g. Would this force not have to be in resonance/harmony/equal too the force in the force equation you hopefully figured out from the paragraph above this one?