A force and its equivalent can be used interchangeably in terms of their effect on an object, following the parallelogram law. Since a force has two main effects—changing the motion of an object (the motion effect) and causing it to deform (the deformation effect)—the question arises: which effect and under what conditions can be considered equivalent?
Physical studies show that:
1. A system of forces acting on a particle can always be replaced by a single equivalent force.
2. A force system acting on a rigid body is generally not equivalent to a single force; instead, it may require both a force and a couple (moment). However, for a concurrent force system (forces meeting at a single point), it can still be represented as a single equivalent force.
3. For non-rigid bodies, where deformation is significant, a force system cannot typically be simplified into a single force.
In summary, only force systems acting on a particle or a concurrent force system on a rigid body can be replaced by a single force. When deformation is involved, regardless of whether the forces are concurrent or not, they cannot be replaced by a single force. This means the parallelogram rule loses its physical meaning in such cases. However, if the focus is solely on the motion effect, and not on deformation, then even non-concurrent forces can be replaced by an equivalent force.
For example, using the parallelogram rule to find the resultant of F1 and F2 in Figure 1 and using this to represent the deformation of a spring caused by these forces is physically meaningless. It would be illogical to say that a person experiencing two equal and opposite pushes feels no force, yet experiences the same deformation. Similarly, two equal and opposite forces applied at different points on a circular plate clearly produce a different effect than if no force were applied.
But why do physics textbooks use experiments, like the one involving rubber bands, to demonstrate the equivalence of forces through deformation? Take the classic parallelogram law experiment as an example. In this setup, a small ball is fixed at a node. The ball is pulled to the same position twice, so the deformation of the rubber bands is identical. According to Hooke’s Law, the restoring force from the rubber band is the same in both cases, denoted as F0. Since the ball remains stationary, the tension F balances F0 when pulled by a spring scale, and similarly, F1 and F2 balance F0 when two scales are used. This shows that F is the resultant of F1 and F2.
The essence of this experiment is to convert a dynamic situation (where the ball's acceleration is hard to measure) into a static one, making it easier to analyze. The equivalence of forces is demonstrated through the consistent deformation of the rubber band, which reflects the balance of forces.
In short, a deep understanding of fundamental concepts related to springs and forces allows us to build a coherent conceptual framework and correctly answer questions about force equivalence.
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