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I'm confused:

In a tug of war game on rough ground who will win? The one who applies greater force on the rope or the one who applies greater force on ground?

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2 Answers 2

up vote 5 down vote accepted

The force on rope is equal for both of them at any time.

For winning the game the force on ground is responsible.

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"The force on rope is equal for both of them at any time." That is not always true. The rope does not move with constant velocity in all cases. –  Ján Lalinský Aug 28 at 5:59
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@bobie, if the rope moved with constant velocity, the forces would be the same too. But this also almost never happens. –  Ján Lalinský Aug 28 at 6:25
    
the loser will the one who will slip..so it totally depends on max static friction which further depends on force applied on ground..force on rope will be same for both unless max static friction limit crosses.. –  Rahul Gupta Aug 28 at 15:19
    
The rope has a very small mass. Therefore the total force on it $F=ma$ is always small, even if $a$ is nonzero. Therefore it's an excellent approximation to say that the forces on the rope are equal. –  Ben Crowell Sep 1 at 16:39

Force on the ground can never be greater than force on the rope

Each players individually exerts some force on the rope and on the ground, but that doesn't means that the same amount of work is done on the rope and on the ground, or that one players exerts the same total force as his opponent. The winner overcomes his adversary only if/when because he exerts an overall greater force distributed both on the rope and on the ground.

Edit:

Looking at one side of the tug of war, there are two forces in line with the rope. The tension in the rope, and the friction force on the ground opposing this force. - Godric Seer

the loser will the one who will slip..so it totally depends on max static friction which further depends on force applied on ground..force on rope will be same for both unless max static friction limit crosses.. – Rahul Gupta

No one must necessarily slip: imagine that your feet rest in a little hole or on a starting block. Ground is just a point of support. The loser will be thrown off balance when he cannot withstand the torque.

, if the players stand at an angle α to the ground the tangential part of the force Fg exerted on the ground..- user9886

the forces are only equal if there is no acceleration (which in the game of tug of war, accelerating yourself and your opponent is the idea). If the force on the rope and the ground are both equal either nothing moves or some other force is being applied in the system. - Godric Seer

The force on the ground can be greater than the force you are exerting on the rope, because this is what causes the acceleration of you backwards. - Godric Seer

A lot of confusion has been brought through the comments adding to what already stated in the accepted answer, which has been soon refuted by Jàn Lalinsky:

the force on rope is equal for both of them at any time. - ramkrishna

Let's se if we can clarify all aspects starting from this crucial one. Suppose a dumbbell of 20 Kg is laying on the floor: g is constant, the reaction of the floor is constant ergo the force on the dumbbell on each direction is equal at any time. Now suppose you lift it with your arm[s]:

enter image description here

g is still constant but your reaction is not constant and the bumbbell will have oscillations whose amplitude depends on your strength. The dumbbell will never be perfectly still, even if the contrary force is constant.

As to the rope the situation is even worse as the opposing force is never equal. Let's consider only two players: the rope will never be still because when one player jerks it, the reaction of the opponent will always be delayed by, say, at least 2/10th of a second and after the jerk he must recover 'hang' from the effort for a fraction of a second or more, so, the reaction of the opponent will restore the position of the centerline or move it further toward the other side.

That as to the first fallacy: the force on rope is never equal for both of them at any time and the rope is never perfectly still.

Now the force on the ground: Each player must always stand at an angle (with the ground) less than 90°, best less than 45°, as in this picture: (http://en.wikipedia.org/wiki/Tug_of_war#mediaviewer/File:Touwtrekken.jpg), with at least one leg fully stretched. (I couldn't find a picture with one player. Consider only the first player: the other players can, in turn, bend knees and relax until the get the 'hang' command:

To achieve this, a person called a "driver" is used to harmonize the team's joint traction power. He moves up and down next to his team pulling on the rope, giving orders to them when to pull and when to rest (called "hanging"). If he spots the opponents tries to pull his team away, he gives a "hang" command, each member will dig into the grass with his/her boots and movement of the rope is limited. When the opponents are played out, he shouts "pull" and rhythmically waves his hat or handkerchief for his team to pull together.

enter image description here

in order to:

  • lower the center of mass and minimize the torque
  • discharge the horizontal force on the resistance of the bones of the legs. If the player lays at 0° and his feet are firmly positioned it is impossible to pull him. That's why touching the ground is prohibited.

It seems obvious that when leg(s) are stretched a player can exert no direct force directly on the ground, as he cannot stretch his leg(s) any further: look at the first player in the picture, how can he exert force on the ground if not by pulling the rope? If he bends his knees, he can exert some force *some*times. On the other hand, if he kept his knees always bent he would be easily carried away. In conclusion, the force on the ground cannot be greater than the force on the rope and cannot be the determining factor to the victory, and moreover no work is ever done on the ground. As prievously stated, ground is only a point of support and he exerts his force mainly on the rope, bending shoulders backwards and forearms closer to the chest.

That as to the second fallacy: the force on the ground cannot be responsible for victory, the victory goes to the player that exerts a greater overall force (on rope/ground) for a longer time. If their strength is equal the one who gets tired first loses.

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This is only true tangential to the rope. So, if the players stand at an angle $\alpha$ to the ground the tangential part of the force $F_{g}$ exerted on the ground is $F_g\cos \alpha$ and therefore the force exerted on the rope is $-F_g\cos \alpha$. Of course I assume here that the rope is long enough and the players are similar enough that the rope can be considered parallel to the ground at all times. –  user9886 Aug 28 at 10:00
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I think this is incorrect, because the forces are only equal if there is no acceleration (which in the game of tug of war, accelerating yourself and your opponent is the idea). If the force on the rope and the ground are both equal either nothing moves or some other force is being applied in the system. –  Godric Seer Aug 28 at 11:51
    
Yes, the two forces on the rope are the same, but I was referring to your statement in the last sentence of your first paragragh. The force on the ground can be greater than the force you are exerting on the rope, because this is what causes the acceleration of you backwards. –  Godric Seer Aug 28 at 16:21
    
@bobie, Looking at one side of the tug of war, there are two forces in line with the rope. The tension in the rope, and the friction force on the ground opposing this force. If the two sides are tied and not moving, then these forces are equal. If one side begins to win, however, they must accelerate their own mass, which means a force imbalance. Since the winner is accelerating their mass in a direction opposite the direction of the tension, the other force must be greater than that tension. –  Godric Seer Aug 29 at 11:03
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@BenCrowell : "Friction is crucial. Your picture of the feet resting in a hole in the ground is not incompatible with describing the force as a frictional force. - Ben Crowell . Crucial for what? for victory? is force frictional when the foot rests in a hole or on a starting block? is force frictional when the dumbbell is resting on the floor and the floor is opposing gravity? The hole is a point of support for the player and a pivot on which to exert a torque for the opponent. –  bobie Sep 2 at 9:08

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