- #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?
In summary, conservative forces depend on position, while non-conservative forces depend on time or velocity. Time varying force F is not conservative because the work done in moving between two points can vary depending on the path taken at different times. This definition of work and energy helps distinguish between fundamental forces and more complex, non-linear forces. To recover the energy put into counteracting a non-conservative force, the object must be moved back to a region where there is a force and then the force can do the work. An example of this is lifting a book from the floor to a table and then pushing it off the table.
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
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Can you see why (i) might be considered a consequence of (ii)?Leo Liu said:
- #3
Physics4Funn
- 25
- 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
- 25
- 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
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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
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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
- 12,276
- 3,470
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
- 25
- 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:
1. Why is a time varying force nonconservative?
A time varying force is nonconservative because it does not follow the principles of conservative forces, which state that the work done by a force should be independent of the path taken. In other words, the work done by a conservative force only depends on the initial and final positions, not the path taken. However, a time varying force changes over time and therefore, the work done by this force can vary depending on the path taken, making it nonconservative.
2. How does a time varying force affect the conservation of energy?
A time varying force violates the principle of conservation of energy because it can change the total mechanical energy of a system. Conservative forces, on the other hand, do not change the total mechanical energy of a system as they only transfer energy between different forms, such as potential and kinetic energy.
3. Can a time varying force be conservative?
No, a time varying force cannot be conservative. As mentioned earlier, conservative forces follow the principle that the work done by the force is independent of the path taken. Since a time varying force does not follow this principle, it cannot be classified as a conservative force.
4. How does a time varying force affect the path of an object?
A time varying force can affect the path of an object by causing it to deviate from a straight line. This is because the work done by a nonconservative force is path-dependent, meaning that the path taken by the object will affect the amount of work done and therefore, the resulting motion of the object.
5. What are some examples of time varying forces?
Examples of time varying forces include friction, air resistance, and drag. These forces change over time as the object moves and can affect the path and speed of the object. Other examples include magnetic and electric forces, which can vary depending on the position and orientation of the objects involved.
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