Complete the analogy : A horse pulling a cart : Motion of cart :: Suspension of a load from spring : ?
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Ambiguity about what body the forces act upon. Failure to subdivide the system into parts for analysis.
Newton's third law action/reaction forces act on different bodies. Forces that contribute to the net force on a body act on that body, not on different bodies.
Failure to define the system that is accelerating.
To analyze this properly, it is probably best to consider the cart and the horse separately. As students are taught in introductory physics courses: first identify and isolate the body that you intend to apply Newton's second law to, then identify all forces acting on that body and only on that body, add them (as vectors) to get the net force, and finally, use Fnet = ma.
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