# Second order differential equation, physics.

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?

• I'm sorry, but no, the dots won't connect. Is anything of what I did above right? Commented Jan 22, 2015 at 16:33
• @fadaes: I should have pointed out two things: 1) your $y$ is with respect to the origin, not equilibrium, and 2) you forgot to include gravity as an external force. You also did not specify your initial conditions as the problem asked. Commented Jan 22, 2015 at 16:36
• So instead of 4.71 there should be 4.71 + (9.81 * 2)? Commented Jan 22, 2015 at 16:40
• @fadaes: I think so, yes. But you would be better served using variables rather than numbers. It is really hard to know what the quantities represent when you just have a bunch of numbers floating about an equation as you do. Commented Jan 22, 2015 at 17:04
• Please point out what numbers you're not sure what means, I'll try to clarify. Commented Jan 22, 2015 at 17:47