Conservative forces
A force is described as a conservative force if the work done by or against it when moving a body between two fixed points depends solely on the initial and final positions of the body and not on the specific path taken. This implies that the work done by or against a conservative force is the same regardless of the path between the initial and final positions.
For instance, gravitational force is a conservative force. To demonstrate this, let’s calculate the work done against gravity when moving a body of mass \( m \) through a height \( AB = h \) using different paths from \( A \) to \( B \).
The given figure depicts the body being lifted vertically. The applied force is \( F = mg \).
As work done \( W = \vec{F} \cdot \vec{S} = FS \cos \theta \)
$\displaystyle \therefore {{W}_{1}}=(mg)\cdot h\cos {{0}^{{}^\circ }}=mgh\text{ }\ldots (1)$
This second figure illustrates the body being moved along a smooth inclined plane \( CB \) of height \( AB = h \) and inclination \( \theta \).
It is evident from from the above figure that the force applied is \( F = mg \sin \theta \), and the displacement along the direction of the force is \( CB \).
Thus,
\[
\text{Work done } = F \cdot CB = F \cdot (CB) \cos 0^\circ
\]
\[
W_2 = mg \sin \theta \times CB = mg \times \frac{AB}{CB} \times CB
\]
\[
W_2 = mgh \text{ }\ldots (2)
\]
The figure below shows the body being lifted through the same height \( AB = h \) over a staircase with \( n \) steps, each of vertical height \( h’ \) and horizontal width \( x \).
$ \displaystyle \begin{array}{l}{{W}_{3}}=n\left[ {mg{h}’\cos {{0}^{\circ }}+mgx\cos {{{90}}^{\circ }}} \right]\\{{W}_{3}}=n\times mg{h}’=mgh\ldots (3)\end{array}$
Again the figure below depicts another situation where the body is being transported through the same height \( AB = h \) using an arbitrary zig zag path. This path can be approximated as a large number of very small horizontal displacements (denoted by \( dx \)) and vertical displacements (denoted by \( dh \)).
Thus, work done
$ \displaystyle \begin{array}{l}=\sum{m}g(dh)\cos {{0}^{{}^\circ }}+\sum{m}g(dx)\cos {{90}^{{}^\circ }}\\{{W}_{4}}=mgh\text{ }\ldots \text{(4)}\end{array}$
From this discussion, we conclude that \( W_1 = W_2 = W_3 = W_4 = mgh \), meaning that the work done remains consistent regardless of the path chosen between the specified initial position \( A \) and final position \( B \). This confirms that gravitational force is indeed a conservative force.
Other examples of conservative forces include:
- Elastic force in a spring.
- Electrostatic force between two charged bodies.
- Magnetic force between two magnetic poles.
The latter two forces are known as central forces because they act along the line joining the centers of two charged/magnetized bodies. Consequently, all central forces are conservative forces.
Properties of Conservative Forces
- Work done by or against a conservative force in moving a body from one position to another is determined only by the initial and final positions of the body.
- Work done by or against a conservative force is independent of the path taken by the body in moving from the initial to the final position.
- Work done by or against a conservative force in moving a body through any closed path (i.e., one where the final position coincides with the initial position) is always zero.
For instance, in the case of gravitational force, if we take work done in moving the body from \( A \) to \( B \) as negative (against gravity), then work done in moving the body from \( B \) to \( A \) (by gravity) must be taken as positive, i.e.,
\[
W_{AB} = -W_{BA}
\]
Therefore,
\[
W_{AB} + W_{BA} = 0 \ldots (5)
\]
In fact, work done in taking the body from \( A \) to \( B \) is stored in the body in the form of potential energy. This energy is expended in moving the body back from \( B \) to \( A \). Therefore, over the complete round trip (\( A \rightarrow B \rightarrow A \)), the total work done is zero.
Non-Conservative Forces
A force is said to be non-conservative if the work done by or against the force when moving a body from one position to another depends on the path taken between these two positions.
For example, frictional forces are non-conservative. If a body is moved from position \( A \) to position \( B \) on a rough surface, the work done against frictional force will depend on the length of the path between \( A \) and \( B \) and not solely on the positions \( A \) and \( B \).
Additionally, if the body is brought back to its initial position \( A \), work has to be done against the frictional forces, which always oppose the motion. Thus, work done against frictional forces in moving the body over a round trip is not zero.
Another example of a non-conservative force is the induction force in a cyclotron. The charged particle returns to its initial position with more kinetic energy than it initially had.
Important:
In a conservative field, work is independent of the path taken. For instance, if a body of mass \( m \) moves with a uniform speed from \( A \) to \( C \) in a gravitational field via path \( AC \) or \( ABC \) (Fig. 4.9), the work done remains the same and is equal to \( mgh \).
Remember:
The work done in moving a body over a smooth inclined plane does not depend on the slope of the inclined plane. \( W = mgh \), and it only depends on the height \( h \) of the inclined plane.
