2018 A/L Physics Past Paper · Question 46 (English Medium)

Tug-of-war — tension in the rope

Two teams start to play tug-of-war using a rope of uniform strength on a hard flat horizontal surface as shown in the figure. Both teams apply equal forces and as a result, the point $O$ on the rope does not move. Consider the following statements made about this situation.

(A) If each of the members of the two groups applies the same force on the rope, the magnitude of the tension throughout the rope is the same.

(B) If the magnitude of tension on the rope exceeds its breaking tension, the rope will break only at a point between $P$ and $Q$ .

(C) The magnitude of the maximum force that can be applied by an individual on the rope depends on the coefficient of static friction between feet of the individual and the surface.

Of the above statements,

(1) only A is true.
(2) only B is true.
(3) only A and B are true.
(4) only B and C are true.
(5) all A, B and C are true.

Answer and explanation

Answer: (4) only B and C are true.

What the question asks

The rope is still. We check three statements about the tension in it.

Relevant theory

When the objects interact (collide), they exert equal and oppsite forces on one another.

The frictional force acting when the surfaces just begin to slip past one another is called the limiting frictional force.

The big idea: the rope does not move. So every piece of it is balanced.

Look at the piece of rope at one person.

The pull on the middle side equals the pull on the outer side plus that person's pull.

So the tension is small at the ends. It is biggest in the middle, between $P$ and $Q$ .

Reading the options

Each option is a mix of A, B and C.

Check each statement on its own. Then pick the option.

Reasoning and calculation

The tension is the same everywhere

Say each person pulls with $F$ . Behind the last person the tension is 0. In front of them it is $F$ . Then $2F$ , $3F$ … and $nF$ between $P$ and $Q$ . So it is not the same everywhere.

Breaks only between $P$ and $Q$

The tension is biggest only between $P$ and $Q$ . The whole pull of both teams is there. The rope is equally strong everywhere, so it breaks there first.

The greatest force depends on $\mu_s$

When you pull the rope, the rope pulls you back. Friction from the ground must hold you. Friction can only be as big as $\mu_s R$ . So the biggest pull depends on $\mu_s$ .

A is false.

Each person adds their pull at their own place.

So the tension changes step by step along the rope.

Why the other options fail

This option says: only A is true.

But A is false.

And B and C are true, but it leaves them out.

So it is wrong.

This option says: only B is true.

B is true. But C is true too, and it leaves C out.

This option says: only A and B are true.

And C is true, but it leaves C out.

This option says: all A, B and C are true.

But A is false. So "all three" cannot be right.

Conclusion

B is true. C is true.

A is false: the tension grows towards the middle.

The answer is option (4).

Explore

The two teams pull the rope with equal forces, so the point $O$ does not move.

Near the last person the rope carries only one person’s pull; towards the middle each person’s pull adds on.

So the tension is largest between $P$ and $Q$ , and that is where the rope would break.

The most one person can pull depends on the friction between their feet and the ground; on slippery ground the feet slide.

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