Newton's Third Law — Question 8

Question 8

Forces at opposite ends of a massive rope are not one third-law pair

A rope of mass 1 kg is pulled right by a hand with 12 N and left by a load with 8 N. Ignore other horizontal forces.

1. Find the rope’s acceleration.

2. Name the partners of the two forces on the rope.

3. Why does their unequal magnitude not violate Newton’s third law?

Question 8 - Solution

Forces at opposite ends of a massive rope are not one third-law pair

(a) Use the rope’s net force.

$$F_{{\mathrm{net}},R}=12-8=4\,\mathrm N\ \Rightarrow\ \boxed{a_R=4/1=4\,\mathrm{m/s^2}}$$ $F_{\mathrm{net}}=12-8=4$ N, hence $a=4$ m/s$^2$ rightward.

(b) Reverse each interface interaction.

$$F_{R\to H}=-12\,\mathrm N,\quad F_{R\to L}=+8\,\mathrm N$$ The rope pulls the hand left with 12 N and pulls the load right with 8 N. Those are the respective partner forces.

(c) Keep the recipients straight.

$$F_{H\to R}+F_{R\to H}=0,\quad F_{L\to R}+F_{R\to L}=0$$ The 12 N and 8 N forces both act on the rope at different interfaces. They are not each other’s third-law partners and may differ to accelerate the rope. Each interface pair separately remains equal and opposite.