For the first problem, I'm not sure your application of N2L is correct. The first step is to assign a coordinate system. I believe z is intended to be the displacement from equilibrium. Do you want z to be positive up or down?
A particle of massm is suspended from a ceiling by a spring of natural length l and
stiffness k. Assume that there is no damping
If the mass is displaced from equilibrium show that the equation governing itsmz'' + kz = 0.
subsequent motion is given by
My working:If we let e be the extension then N2L is ke-mg=0 so ke=mg.
An engineer measures the mass of the particle to bem = 2, and furthermore k = 4.
He also finds that internal friction in the spring provides damping that is proportional
to the velocity of the particle with a constant of proportionality μ = 4.
Show that the equation for the subsequent motion after the displacement of the2z''+ 4z' + 4z = 0
mass is now given by
My working: what is the constant of proportionality?
Much closer. However, there are still two problems.
1. z is measured from the equilibrium point of the mass on the spring, whereas the equation you wrote is really more correct for a different variable y, measured from the equilibrium point of the spring without a mass on it. This point is getting at the heart of the problem here: what's the difference between the system with a mass and without a mass? Another way of phrasing it is this: what is the effect of gravity on such a vertical mass-spring system?
2. The kz term should have the opposite sign (in addition to the change mentioned above), since the spring is a restoring force.
Compare these two scenarios: the spring without any mass in equilibrium, with the mass on the spring in equilibrium. Just imagine hanging the mass on the spring, and waiting until it stops bobbing. As a thought experiment, what's the result going to be?