# Spring mass system [duplicate]

Need some clearance on a couple of things, here is a problem in it's complete form,

A mass of $400 \textrm{ g}$ stretches a spring by $5 \textrm{ cm}$.

(a) Find the spring constant $k$, the angular frequency $\omega$, as well as the period $T$ and frequency $f$ of the free undamped motion for this spring-mass system.

(b) Find the general solution $x(t)$ of the DE for the free spring-mass system.

(c) Suppose that an exterior force of $F(t) = 27\sin(13t) \textrm{ N}$ acts on the spring-mass system. Find the equation of motion of the system if the mass initially is at rest in its equilibrium position.

I have the general solution $x(t) = c_1\sin(14t) + c_2\cos(14t)$. What initial conditions do I use here to find the constants? $x(0) = 0, x'(0) = 0$?

For part (c), I just set the force equal to my general solution and use the method of undetermined coefficients, right? I'm confused at this point as to what my guess should be.

## marked as duplicate by abel, Jonas Meyer, user147263, JMoravitz, N. F. TaussigMar 10 '15 at 2:41

• Hint: You need to set up the differential equation first. – Dylan Mar 10 '15 at 2:23

For the mass spring system, you have the general differential equation:

$$m\frac{d^2 x}{dt^2} + b\frac{dx}{dt} + kx = F(t)$$

where $F(t)$ is the external force and $b$ is the friction coefficient. Since this system is umdamped, $b = 0$.

If you want to find the initial conditions, look at how the spring behaves at $t = 0$. At what location is the mass? Is it moving or at rest?

For part (a), $F(t) = 0$ so you have the homogeneous equation

$$\frac{d^2 x}{dt^2} = -\omega^2 x$$

where $\omega = 14$

For part (b), set $F(t) = 27\sin(13t)$, and you'll have to solve this equation

$$m\frac{d^2 x}{dt^2} + kx = 27\sin(13t)$$

with the same initial conditions