# Problem of harmonic motion

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1. Jun 29, 2016

### Cozma Alex

1. The problem statement, all variables and given/known data
A particle with mass m is undergoing with harmonic motion with a period T, we introduce an external force F proportional to velocity v so that F= -bv with b a constant and we assume that the particle continues to oscillate how does the period change?

2. Relevant equations F= m a ; v'= a; x'= v; x''= a

3. The attempt at a solution

So my idea was creating the differential equation of the motion:

Before the external force to be applied :

m x" + k x= 0 (there, must be a force F=-kx even if the problem doesn't mention it so that the harmonic motion exists before the application of the external force) so T= 2 pi (m/k)^0.5

After the force:

mx'' + b x' + k x= 0 , the problem is that this equation has not as a solution a function like this x (t)= A cos ( wt + phi) but a linear combination of exponential function so I can't figure out what the period is... (to solve the equation is used wolfram alpha and it doesn't give me a sinusoidal function)

2. Jun 29, 2016

### blue_leaf77

It's an elementary problem in differential equation, called damped harmonic motion. See this link about a way to solve such kind of problem. Note that since the problem assumes the particle to continue undergoing oscillation, this problem implies the underdamped solution.

3. Jun 30, 2016

### Cozma Alex

The relation in the link doesn't give me info about the new period of oscillation, because the solution is not a sinusoidal function but expo, how do I get it?

4. Jun 30, 2016

### Cozma Alex

Ok, i found it, thanks