# Forced Oscillation with complex numbers

1. Nov 27, 2015

### namesis

1. The problem statement, all variables and given/known data
If a force $$F = F_0 cos (\omega t) = \Re{\{F_0 e^{i \omega t}\}}$$ is applied to a body of mass m attached to a spring of constant k, and $$x = \Re\{z\}$$. Show that the following equation holds:
$$m \ddot{z} = - k z + Fe^{i \omega t}$$ .

2. Relevant equations
Newton's second law.

3. The attempt at a solution
I tried to solve the problem assuming that $$z = x + y i$$ with x and y being real numbers and show that the real parts of the two sides of the equation are equal and so are the imaginary parts. However, although for the real parts we just get Newton's second law, I could not anything useful by the imaginary parts.

2. Nov 27, 2015

### BvU

Hello Namesis,

Don't you get Newton's law for the imaginary part as well ?

3. Nov 28, 2015

### nrqed

The key point is that only the real part has physical meaning, you may ignore completely the imaginary part. The physical equation is the real part of $m \ddot{z} = - k z + Fe^{i \omega t}$, the imaginary part can simply be discarded, it has no meaning (as is clear from the fact that the y you introduced in $z= x + i y$ has not been assigned any physical meaning.

4. Nov 28, 2015

### namesis

I see now. My point was that you can perfectly write $x$ as $x + 2342 i \in \mathbb{Z}$ without that obeying the equation. So every $z$ that is a solution of the equation, has the property $x = \Re{\{z\}}$ but not every $z$ with the previous property satisfies the equation. The problem was stated in a way that it implied that what I had to show is: "Show that $x = \Re{\{z\}}$ if and only if $z$ satisfies the equation."