# Pulling the rope with one hand is as heavy as with two hands?

There is this box, having two holes, and the ropes going through each of them.
I try to pull the left rope with my left hand... And it's heavy...
Then I try to pull the right rope with my right hand... It's also heavy, as heavy as before.

Ok so... I think both ropes are directly connected to a single weight.
So I try to pull both ropes with my both hands, hoping to reduce the force exerted by each arm by $$50\%$$...
But... I'm wrong... The box is no easier to lift with two arms!

Ah, of course. There are two weights. Each rope is connected to different weight...
Yes, that makes sense!

Let's open this box to prove that I'm right.

...

Wait... What?! There is only one weight there?! Wow, what an amazing contraption!

So, do you know how can this be happening?

• Do you measure how hard it is to lift the weight or to keep it in a high position, or both? – Arnaud Mortier Apr 26 at 10:30
• @noedne Ten people can lift a small car, one person cannot. – Arnaud Mortier Apr 26 at 10:31
• @noedne Given the final contraption, it's expected to behave as the story. – athin Apr 26 at 10:34
• @ArnaudMortier The measurement can be exact. Say, you need $x$ Newton to perform those $3$ pullings. (Well, for the third one, it should be $2x$ Newton in total as using both of hands $= x + x$.) – athin Apr 26 at 10:35
• @athin OK, so what's actually happening is that the box weighs twice as much (not the same amount) when pulling both ropes compared to when pulling with either rope individually. Because the weight of the box doubles when using both arms, the force exerted by a single arm is identical in all 3 cases (left arm, right arm, both arms). – Nuclear Wang Apr 26 at 15:04

The weight is attached to a pulley. The two ends are of the same rope that runs through the pulley. Also, the rope is tied to two ropes that connect it to the top of the box on either side of the pulley. When you pull one end of the rope, the pulley lets you cut the weight in half, but when you pull on both ends, you do not use the pulley and lift the entire weight.

Below is my attempt at depicting this contraption.

• My physics is quite rusty, so please let me know if this is mistaken. – noedne Apr 26 at 10:45
• All hail the mighty man in the box, balancing a table on his long arms! – infinitezero Apr 26 at 11:34
• @HansJanssen Or are you suggesting that when you pull one end of the rope, you are still holding the other end taut? – noedne Apr 26 at 13:07
• And yeah, actually this is the trick! >< – athin Apr 26 at 14:01
• @infinitezero This can't be unseen. – user1717828 Apr 26 at 16:09

Maybe the ropes are

attached to different ends of the thin weight, which is lying flat inside the box.

When you pull only one rope,

only that end of the weight rises. The centre of mass of the weight moves up by half the distance pulled, so the force required is half of the weight of the weight

And when you pull on both ropes

The whole weight rises, and the force is equally split between the two ropes.

In both cases, the force on any pulled rope is the same, namely

one half of the weight of the weight.

• My physics is rusty, but I don't think the distance moved affects the forces. – noedne Apr 26 at 11:10
• @noedne Sure it does. It all comes back to the law of the lever: gravity pulls down at the center of mass, and the rope pulls up at exactly twice the distance from the fulcrum, which we know from the fact that the rope moves twice as much as the center of gravity. – Bass Apr 26 at 11:23
• This is also a good (alternative) answer. I should mention that the weight is not thin, it's a large solid one. – athin Apr 26 at 14:00
• Well, if you allow the weight to be suspended by the ropes (as it would seem from the answer with the tick), you can have a horizontal rod with the ropes attached symmetrically around the centre, from which the weight is hanging. No pulleys required. – Bass Apr 26 at 14:35
• @Bass I think the pulley is needed for the situation to hold while lifting; the rod would work to balance the ropes at one specific height, but at any other height one of the ropes would be slack, so the weight would not be distributed. – noedne Apr 26 at 15:01

The ropes themselves are significantly heavier than the single weight so that when you lift with either with the left or right hand you're mostly lifting the rope on that side (together with small contributions from the weight and the end of the other rope). When you lift with both hands, you are lifting both ropes.

• That only works if the weight would be 0 kg. If that would be the case it's not really called a weight right... – Hans Janssen Apr 26 at 13:05
• @HansJanssen Maybe there's a weight in the box but not connected to the ropes? – SpoonMeiser Apr 26 at 13:33
• This is actually a good (alternative) answer. Sorry but I should say that the rope is a usual rope which is not heavy, and certainly, the weight should be much heavier than the rope. – athin Apr 26 at 13:59
• @HansJanssen or there is simply not a weight. Just heavy ropes. – abligh Apr 26 at 19:19
• @abligh Not a weight seems a very unlikely explanation to the question why "There is only one weight there"... – Hans Janssen Apr 29 at 9:00

Maybe

There is a pulley hinged at the box ceiling. There is a single rope. The weight has two pulleys attached to it for the rope to pass through. Both ends of the rope are tied to a ceiling outside the box. The rope, from one side, enters the box, passes under the first pulley of the weight, over the hinged pulley, down under the second pulley attached to the weight and then out of the box all the way to the other tied end. When you pull either side of the rope (that appears as a separate rope), the mechanical advantage is 4 (thanks to Hans Janssen for correcting me here), thus reducing the effective weight by 4. When you pull at both sides, the hinged pulley's function is lost and you end up with the mechanical advantage of 2 for each hand. Consequently, the effective weight does not change.

• Actually, with that design the mechanical advantage when pulling one rope is 4, and it is 2 when pulling both ropes. Still solves the puzzle tough ;) – Hans Janssen Apr 26 at 13:14
• Yep, this is also how to solve the puzzle with pulley trick. Sadly @noedne is faster here >< – athin Apr 26 at 14:05

This explains it:

The box is sealed from outside and the ropes are not connected to the weight. When you pull the ropes out you are creating a pressure difference caused by the displacement of the volume occupied by the ropes (isochoric process). If the ropes are equal in size, this will mean that you will get the same displacement per rope, i.e. the same pressure difference per rope thus the same force per rope.

This is analogous to a case where you connect two syringes (instead of ropes) to a sealed box (optionally with something just laying inside it).

• Ah, this will work if the holes are small enough only for the rope, and no air can come from outside, cmiiw? – athin Apr 26 at 14:03

Maybe

The ropes are not attached to the weight, but they themselves are quite heavy.

Thinking outside the box:

The weight could be lying flat on the bottom of the box. The ropes are twisted around a pulley or edge to the left rope actually drags the weight to the left, and the right rope drags the weight to the right. The weight and box are chosen so the force needed to drag the weight is half of the force needed to lift the box. To when one rope end is dragged, a certain force is required. When both ropes are pulled, the weight cannot be dragged and the whole box is lifted, using twice the force.

Here is a drawing of the setup:

As an alternative to my previous answer, this would also work:

The ropes are independent but connected to the weight. But if the ropes are not under tension, and just rolled down on the base of box, when you pull the ropes up you would just feel the weight of the ropes, not the weight. Thus, pulling one, the other, or both, will mean that you will be just lifting independent (but identical) ropes out of the ground.

• Ah, I should mention that the rope can be pulled, means it's not stuck to the base or any kinds of stuff which make it unable to move. – athin Apr 26 at 14:08