I need your input on this exercise I'm doing:
"A 2-kg mass is suspended from a string. The displacement of the spring-mass equilibrium from the spring equilibrium is measured to be 50 cm. If the mass is displaced 12 cm downward from its spring-mass equilibrium and released from rest, set up the initial value problem if no damping is present."
The spring constant is:$$k = \frac{2kg * 9.81m/s}{0.5m} = 39.24N/m$$
The additional force is $$39.24N/m * 0.12m = 4.71N$$
Which gives the differential equation: $$2y'' + 39.24y = 4.71$$
Is this correct?
Newton's 2nd Law: The sum of the forces equal $m a$.
Sum of the forces:
$$\underbrace{-k y}_{\text{spring force}} + \underbrace{m g}_{gravity}$$
Here, $y$ represents displacement from equilibrium. Obviously, downward is positive here. Note the minus sign on the spring force is there because the spring force opposes the direction of motion.
Initial conditions:
$$y(0) = 12 \quad \dot{y}(0) = 0$$
Can you take it from here?