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RJLiberator
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Homework Statement
Solve for the steady-statem otion of a forced oscillator if the forcing is F_0*sin(wt) instead of F_0*cos(wt). Use complex representations.
Homework Equations
e^(i*x) = cos(x)+isin(x)
The Attempt at a Solution
I assume, First, steady-state means without damping.
Next, we have the equation
m*x'' = -kx + F_0*sin(wt)
where x'' is the second derivative of x position with respect to time. Also known as acceleration.
So, we have:
z'' + w_0^2*z = F_0/m * (-i*e^(iwt)) where Re(z) = x
So far so good? I think I changed it correctly, normally when you do F_0*cos(wt) as forcing, you just switch it to e^(iwt)
Now, if everything is looking good up to here, we take z = A*(-ie^[i(wt-φ)])
After taking the second derivative of this and inputting it into our equation, our equation looks like:
i*w^2*A*e^[i(wt-φ)]-w_0^2*A*i*e^[i(wt-φ)] = F_0/m * (-i*e^(iwt))
dividing through by -i*e^(iwt) nets us
F_0/m = A(w_0^2 - w^2)*e^(-iφ)
Simplifying: F_0/m *e^(iφ) = A(w_0^2-w^2)
But now I seem to have run into a problem. The real part of this is F_0/m*cos(φ)
But this is the exact same solution to the forcing with F_0*cos(wt).
I thought the problem wants me to reach Re(z) = F_0*sin(wt)
Any help? Did I go wrong somewhere?