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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. Taussig Mar 10 '15 at 2:41

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  • $\begingroup$ Hint: You need to set up the differential equation first. $\endgroup$ – 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


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