# Problem of harmonic motion

physics user1

## Homework Statement

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?

## Homework Equations

F= m a ; v'= a; x'= v; x''= a[/B]

## 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[/B]

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)

blue_leaf77
Homework Helper
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.

• physics user1
physics user1
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.

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?

physics user1
Ok, i found it, thanks