Find dissipative function for the non-linear force f=-bv^n

In summary, the dissipative function for a nonconservative force f=-bvⁿ in Lagrangian, where v is the velocity, is D=-(1/2)bv². This is because f=∂D/∂x and the dissipative function should be D=-(1/(n+1))bv^(1+n) according to f=∂D/∂x.
  • #1
mcconnellmelany
21
5
Homework Statement
For a nonconservative force,
What would be the dissipative function for a force f=-bvⁿ in Lagrangian
(Where v is the velocity)
[#qoute for a nonconservative force f=-bv
The dissipative function is D=-(1/2)bv² ]
Relevant Equations
##\frac{d}{dt}(\frac{\partial L}{\partial \dot x})=\frac{\partial L}{\partial x} - \frac{\partial F}{\partial \dot x}##
For a nonconservative force,
What would be the dissipative function for a force f=-bvⁿ in Lagrangian
(Where v is the velocity)
[#qoute for a nonconservative force f=-bv
The dissipative function is D=-(1/2)bv² ]

Since ##f=\frac{\partial D}{\partial \dot x}## so the dissipative function should be ##D=-\frac{1}{n+1}bv^{1+n}##, isn't it?
 
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  • #2
mcconnellmelany said:
Since ##f=\frac{\partial D}{\partial \ddot x}##
Do you mean ##f=\frac{\partial D}{\partial \dot x}## ?
 
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  • #3
haruspex said:
Do you mean ##f=\frac{\partial D}{\partial \dot x}## ?
Hmm! It was a typo.
 
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  • #4
mcconnellmelany said:
Hmm! It was a typo.
Your answer looks right to me.
 

1. What is a dissipative function?

A dissipative function is a mathematical representation of energy dissipation in a system. It describes the loss of energy due to friction, resistance, or other non-conservative forces.

2. How do you find the dissipative function for a non-linear force?

To find the dissipative function for a non-linear force, you need to integrate the force function over the displacement of the system. This will give you the total work done by the force, which can then be used to calculate the energy dissipated.

3. What is the significance of the coefficient b in the force function f=-bv^n?

The coefficient b represents the strength of the dissipative force. A higher value of b indicates a stronger dissipative force, which leads to more energy dissipation in the system.

4. How does the exponent n affect the dissipative function?

The exponent n determines the nature of the non-linear force and how it changes with displacement. A higher value of n indicates a more rapid increase in force with displacement, resulting in a greater amount of energy dissipation.

5. Can the dissipative function be negative?

Yes, the dissipative function can be negative if the force and displacement are in opposite directions. This indicates that the dissipative force is doing work on the system, resulting in a decrease in energy rather than dissipation.

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