Question

a rope is attached to two heavy chain links that are connected to each other, as shown. the rope is pulling the chain links up at a constant speed. the top chain link exerts a force of magnitude ( f_{mathrm{tb}} ) on the bottom link, and the bottom link exerts a force of magnitude ( f_{mathrm{bt}} ) on the top link. which of the following correctly compares ( f_{mathrm{tb}} ) and ( f_{mathrm{bt}} ) and provides a valid justification?

a ( f_{mathrm{tb}}>f_{mathrm{bt}} ), because the chain links are moving upward.
b ( f_{mathrm{tb}}>f_{mathrm{bt}} ), because the rope only exerts a force on the top link.
c ( f_{mathrm{tb}}=f_{mathrm{bt}} ), because the chain links are moving at a constant speed.
d ( f_{mathrm{tb}}=f_{mathrm{bt}} ), because the chain links are an interacting newtons third law pair.

Explanation:

Newton's third law states that for every action, there is an equal and opposite reaction. When two objects interact, the forces they exert on each other are equal in magnitude and opposite in direction. Here, the top chain link exerting a force \(F_{TB}\) on the bottom link and the bottom link exerting a force \(F_{BT}\) on the top link are an action - reaction pair (Newton's third law pair). The fact that the chain links are moving at a constant speed (which would imply zero net force on the system as a whole, from Newton's first law \(F_{net}=ma\) with \(a = 0\)) is not the reason for \(F_{TB}=F_{BT}\). The reason is the fundamental nature of their interaction as per Newton's third law.

• Option A: Incorrect. Motion (upward in this case) does not determine the equality of action - reaction forces. Newton's third law is about interaction, not motion.
• Option B: Incorrect. The fact that the rope pulls only the top link is not relevant to the equality of \(F_{TB}\) and \(F_{BT}\). These two forces are between the two chain links themselves.
• Option C: Incorrect. Constant speed (zero acceleration) implies \(F_{net}=0\) for the system (from \(F = ma\)), but this is not the reason for \(F_{TB}=F_{BT}\). \(F_{TB}\) and \(F_{BT}\) are equal because of Newton's third law, not because of the motion state (constant speed).
• Option D: Correct. Since \(F_{TB}\) and \(F_{BT}\) are an interacting Newton's third law pair, by Newton's third law \(F_{TB}=F_{BT}\).

Answer:

D. \(F_{TB}=F_{BT}\), because the chain links are an interacting Newton's third law pair.