Spring DE- Find position at any time t.

[MLB32]

Thanks for the help.

Homework Statement

A spring is stretched 12 cm by a force equal to 360 dynes. A mass of 120g is attached to the end of the spring. The system is then set to motion by pulling the mass 3cm above the point of equilibrium and releasing it, at time t=0, with initial speed of (3/2) cm/sec directed downward. Find a) the position of the mass at any time t, and b) When will the mass return to the equilibrium position for the first time? Why?

Homework Equations

The Attempt at a Solution

k=360/12=30
120y'+30y=0
λ^2+1/4=0
λ= +/- .5i
... STUCK
Find a) the position of the mass at any time t, and b) When will the mass return to the equilibrium position for the first time? Why?

 

[Apphysicist]

Without knowing the relation of dynes to Newtons off the top of my head, let me write out your DE using variables (I assume this is not a simpler case of horizontal motion, rather, by saying "above" and "directed down" you are dealing with a vertical spring system so the force of gravity constantly acts):

mx''=mg-kx is simply the expression of Newton's Second Law (with a coordinate system that has the +y-direction oriented downward)

mx''+kx=mg

This is a nonhomogeneous second order DE which can be solved by adding the particular solution and homogeneous solutions. The homogeneous solution is found by the way you started, by assuming mx''-kx=0. The particular solution can be solved by using undetermined coefficients (in this case, let your particular solution x(t)=A, since the right hand side is a constant with respect to time).

x(t)-particular...x=A, x'=0, x''=0, then substitute to find x(t)-particular.

You will add this term to the homogeneous solution which started to attempt to solve.

λ= +/- i*sqrt(k/m) , which is what you, just check on your units.

x(t)-homogeneous...you work toward a solution that looks like x(t)=C*e^(λt)...of course you have two λs and so x(t)-homogeneous is the linear combination of your two solutions, C is a constant that depends on your initial conditions. Use euler's formula:

e^(iB)=cos(B)+isin(B), B a real value...when you do the linear combination, without proving it here, you can just say C_1*cos(B)+C_2*sin(B), dropping the i.

That should be plenty of a start for you, good luck.