- #1
Leo Liu
- 353
- 156
Can anyone please tell me why time varying force F is not conservative? That is, what makes a force not depending on the position nonconservative?
Why is a time varying force nonconservative?
Click For Summary
Discussion Overview
The discussion centers around the nature of time-varying forces and their classification as non-conservative forces. Participants explore the implications of forces that depend on time and velocity, examining how these characteristics affect the work done and energy conservation in various contexts.
Discussion Character
- Exploratory
- Debate/contested
- Technical explanation
- Conceptual clarification
Main Points Raised
- Some participants suggest that a force depending on time can result in different work done when moving between two points at different times.
- Others argue that forces depending on velocity can also lead to different work outcomes based on the speed of movement along different paths.
- A participant notes that non-conservative forces arise from complex interactions, citing friction and viscosity as examples influenced by various factors.
- One participant mentions that for a field to be conservative, the curl must equal zero, but questions the appropriateness of calling a time-dependent field 'conservative' in a physics context.
- Another participant raises the issue of energy recovery when work is done against a force that subsequently disappears, questioning the feasibility of recovering that energy.
- Some participants discuss the historical separation of time and space in the context of conservative forces, suggesting that modern understandings complicate these concepts.
- Examples are provided, such as lifting a book against gravity, to illustrate the dynamics of work and energy in conservative versus non-conservative scenarios.
Areas of Agreement / Disagreement
Participants express differing views on the implications of time-dependent forces and their classification as non-conservative. There is no consensus on the nature of energy recovery in scenarios involving disappearing forces, and the discussion remains unresolved regarding the broader implications of time and velocity on conservative forces.
Contextual Notes
Participants highlight the complexity of defining conservative forces in the context of time-dependent fields and the historical evolution of these concepts. The discussion reveals limitations in understanding energy recovery and the conditions under which forces can be considered conservative or non-conservative.
Can anyone please tell me why time varying force F is not conservative? That is, what makes a force not depending on the position nonconservative?
- #2
- 29,616
- 21,427
Can you see why (i) might be considered a consequence of (ii)?Leo Liu said:
- #3
Physics4Funn
- 24
- 6
If a force depends on time, then the work done in moving between two points can be different, especially if that path is taken at different times.
If a force depends on velocity, similarly, take two different paths at two different speeds, then the work is different.
From the history of the concept of work and energy, this very strict definition helped tell the difference between a fundamental force, electric, gravity, from a force that they had no way to precisely describe that force. E.G. friction, viscosity. Non-conservative forces come from complex, non-linear interactions between objects. Friction comes from: applied normal force, surface roughness, chemical composition of objects. Viscosity comes from highly non-linear Navier-Stokes fluid equations.
Our understanding of the universe would be very different without this definition.
If a force depends on velocity, similarly, take two different paths at two different speeds, then the work is different.
From the history of the concept of work and energy, this very strict definition helped tell the difference between a fundamental force, electric, gravity, from a force that they had no way to precisely describe that force. E.G. friction, viscosity. Non-conservative forces come from complex, non-linear interactions between objects. Friction comes from: applied normal force, surface roughness, chemical composition of objects. Viscosity comes from highly non-linear Navier-Stokes fluid equations.
Our understanding of the universe would be very different without this definition.
- #4
Physics4Funn
- 24
- 6
1) and 2), time and velocity are definitely related: v=distance/time. But, especially at the time of creation of the concept of conservative force, time and space were considered completely separate and different. Now that we know that time and space are related, then whole subject can become more complicated.PeroK said:
I always keep specific examples of phenomenon in mind to keep my understanding specific.
- #5
etotheipi
For a field ##\vec{F}## to be conservative, you just need that ##\nabla \times \vec{F} = \vec{0}##. This can still be satisfied with a time-dependent field, however it would be strange to call such a field 'conservative' in a physics sense.
Mainly because, suppose a particle moves under a force ##\vec{F} = \vec{F}(\vec{r}, t)##, which satisfies ##\nabla \times \vec{F}(\vec{r}, t) = \vec{0} \implies \vec{F}(\vec{r}, t) = - \nabla \phi(\vec{r}, t)##. Then$$\vec{F} = -\nabla \phi = m\ddot{\vec{r}}$$ $$m\ddot{\vec{r}} \cdot \dot{\vec{r}} + \nabla \phi \cdot \dot{\vec{r}} = \vec{0}$$Chain rule tells you that$$\frac{d\phi}{dt} = \nabla \phi \cdot \dot{\vec{r}} + \frac{\partial \phi}{\partial t}$$Substitute for ##\nabla \phi \cdot \dot{\vec{r}}## in the previous expression (and noting that ##v^2 = \dot{\vec{r}} \cdot \dot{\vec{r}}##),$$\frac{d}{dt} \left (\frac{1}{2}mv^2 + \phi \right) = \frac{\partial \phi}{\partial t}$$You can see that the energy measure ##E = \frac{1}{2}mv^2 + \phi## is not conserved if the potential is time-dependent.
Mainly because, suppose a particle moves under a force ##\vec{F} = \vec{F}(\vec{r}, t)##, which satisfies ##\nabla \times \vec{F}(\vec{r}, t) = \vec{0} \implies \vec{F}(\vec{r}, t) = - \nabla \phi(\vec{r}, t)##. Then$$\vec{F} = -\nabla \phi = m\ddot{\vec{r}}$$ $$m\ddot{\vec{r}} \cdot \dot{\vec{r}} + \nabla \phi \cdot \dot{\vec{r}} = \vec{0}$$Chain rule tells you that$$\frac{d\phi}{dt} = \nabla \phi \cdot \dot{\vec{r}} + \frac{\partial \phi}{\partial t}$$Substitute for ##\nabla \phi \cdot \dot{\vec{r}}## in the previous expression (and noting that ##v^2 = \dot{\vec{r}} \cdot \dot{\vec{r}}##),$$\frac{d}{dt} \left (\frac{1}{2}mv^2 + \phi \right) = \frac{\partial \phi}{\partial t}$$You can see that the energy measure ##E = \frac{1}{2}mv^2 + \phi## is not conserved if the potential is time-dependent.
- #6
A.T.
Science Advisor
- 13,263
- 4,184
If you do work against a force, and then the force disappears, how could you recover the energy you put in?Leo Liu said:
- #7
- 29,616
- 21,427
Thanks for that, but my question was addressed to the OP.Physics4Funn said:
- #8
Leo Liu
- 353
- 156
Yet how could you make a pair of forces disappear?A.T. said:
Also, I would like to know how you can recover the energy after counteracting a force to move an object to a region in which the force and its paired force disappear.
- #9
A.T.
Science Advisor
- 13,263
- 4,184
Make it time dependent.Leo Liu said:
Move it back where there is a force, then let the force do the work.Leo Liu said:
- #10
Physics4Funn
- 24
- 6
An example: a book on the floor. No net force: a pair of forces that seem to disappear. Gravity pulling down. Floor pushing up. Lift the book to the table: work is done on the book. On the table, again no net force: gravity pulling down, table pushing up. Since gravity is conservative, then extract that work by pushing the book off the table and gravity pulls it down returning that work of lifting as kinetic energy.Leo Liu said:
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