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phydev
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Why elementary work is defined as δW=Fdr?
My ques. is not on the definition; it is on why it cannot be dW=Fdr?
My ques. is not on the definition; it is on why it cannot be dW=Fdr?
tommyli said:It is only an exact differential int he case of a conservative force. Precisely, if there is a potential we write
dU = -Fdr
which implies that F = -dU/dr.
D H said:Work is a line integral:
[tex]W=\oint \mathbf F \cdot d\mathbf r[/tex]
It is path independent in the case of a conservative force, but in general the integral depends on the path. In other words, [itex]\mathbf F \cdot d\mathbf r[/itex] is not an exact differential by definition (an exact differential is path independent).
phydev said:I'm not talking about U(potential energy function), I'm asking about W.
I know that in case of conservative/potential field δW=-dU.
Reference: Fundamental Laws of Mechanics, IE irodov
from equation 3.1 to 3.49
wherever needed he used δA for elementary work, in general!
tommyli said:Whenever you write df it implies that the differential operator is applied to the function f, so if you write work as an exact differentail dW implicitly you are saying force can be written as the derivative of some function of spatial coordinates.
phydev said:Yeah! right!
Now, what does it further imply?
Cannot force be derivative of a function of spatial coordinates?
I think I have got it, but request you to elaborate so that I may confirm.
Thanks!
tommyli said:If force is a derivative of some function of spatial coordinates, this function is called the potential energy, and the force is conservative. Non-conservative forces are certainly not derivatives of any function.
Elementary work is not an exact differential because it depends on the path taken between two points in a thermodynamic system. This means that the work done on or by the system will be different depending on the specific path taken, even if the starting and ending points are the same. In other words, the value of work is not solely determined by the state of the system but also by the process used to change the system's state.
In thermodynamics, an exact differential means that the value of a property only depends on the initial and final states of the system, and not on the path taken between these states. This is in contrast to non-exact differentials, such as elementary work, which depend on the specific path taken in changing the system's state.
The first law of thermodynamics states that energy cannot be created or destroyed, only transferred or converted. This means that the total change in energy of a system is independent of the path taken, and therefore energy is an exact differential. However, work, which is a form of energy transfer, is not an exact differential as it depends on the path taken between two states of the system.
An example of a process where elementary work is not an exact differential is the expansion of a gas against a constant external pressure. If the gas expands slowly and reversibly, the work done is equal to the external pressure multiplied by the change in volume. However, if the gas expands rapidly and irreversibly, the work done will be greater than the external pressure multiplied by the change in volume. This is because the rapid expansion process involves dissipative forces, such as friction, that contribute to the work done.
In thermodynamic calculations, exact differentials are used to determine the total change in a property of a system, such as internal energy or entropy, by integrating over the entire path between the initial and final states. This allows for the calculation of thermodynamic quantities that are independent of the specific path taken, such as the change in internal energy of a system. In contrast, non-exact differentials, like elementary work, must be calculated separately for each specific path and cannot be integrated to determine a total change in a property.