# Damped harmonic motion

## Homework Statement

Reading chapter 4 of Morin's "Introduction to classical mechanics" I came across to the explanation of the damped harmonic motion.

The mass m is subject to a drag force proportional to its velocity, ##F_f = -bv ##.
He says that the total force of the mass is ##F= -b \dot{x} -kx## and considering that ##F= m\ddot{x} ## we get this differential equation $$\ddot{x} +\frac{b}{m} \dot{x} + \frac{k}{m} x=0$$

But the total force should be ##F= -b \dot{x} +kx##, shouldn't it? These two forces have opposite direction!

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But the total force should be F=−b˙x+kxF=−bx˙+kxF= -b \dot{x} +kx, shouldn't it? These two forces have opposite direction!

if x is extension or compression in the spring the force experienced is always opposite to the
direction of change in x so spring force must be -k.x

Orodruin
Staff Emeritus
Homework Helper
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Also, ##x## is not the same as ##\dot x##. Depending on their relative sign, the drag and restoring force may or may not be in the same direction.

Also, xxx is not the same as ˙xx˙\dot x. Depending on their relative sign, the drag and restoring force may or may not be in the same direction.

however the velocity /motion of the body will always be opposed by the drag so it should be -damping coefficient times the velocity.
take two cases...

1. .x is increasing ..extension..so -k.x and velocity is in the same direction so again the drag will be opposite

2. take compression x is reducing so so restoring force will oppose compression and velocity is in the direction of compression so drag will be outward.

Am i right...

Orodruin
Staff Emeritus