Is the Total Force in Damped Harmonic Motion Always Opposite to Velocity?

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Homework Help Overview

The discussion revolves around the concept of damped harmonic motion as presented in a classical mechanics context. The original poster questions the formulation of the total force acting on a mass subject to both a damping force and a restoring force from a spring.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants explore the relationship between the damping force and the spring force, questioning the signs and directions of these forces. They discuss the implications of velocity and displacement on the forces involved in the system.

Discussion Status

The discussion is active, with participants offering insights into the nature of the forces at play and how they relate to the motion of the mass. There is an exploration of different scenarios regarding the direction of displacement and velocity, leading to a nuanced understanding of the forces involved.

Contextual Notes

Participants are examining the definitions and assumptions regarding the forces in damped harmonic motion, particularly focusing on the signs of the forces and their directions relative to velocity and displacement. There is an acknowledgment of the complexity introduced by the relative signs of displacement and velocity.

fcoulomb
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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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fcoulomb said:
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
 
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.
 
Orodruin said:
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...
 
drvrm said:
1. .x is increasing ..extension..so -k.x and velocity is in the same direction so again the drag will be opposite
This depends on what you mean with "x is increasing". If you assume positive ##x##, yes. It would be more accurate to say that the forces act in the same direction whenever you move away from the equilibrium position and in opposite directions whenever you move towards the equilibrium position.
 
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