This can be considered a follow-up turn-around question for this similar question.

I recently came across this interesting analysis of the game of tug of war.
One of the closing discussions is the following image:

Sample image

However, the writer follows up that:

[Given there are multiple people on both teams,] the drawing above shows that the shorter players on the left team also experience an upward component of force, and therefore the friction on their feet is reduced. The shorter player on the right team certainly experiences decreased friction, but the other players on the right team have increased friction at their feet.

How do all the forces add up? Is it beneficial to put taller people first in a game of tug of war as shown in the diagram above, or do the forces cancel each other out?

  • $\begingroup$ Note: if you play the game properly (i.e. with a muddy pond between the teams) then the drawing is not to scale. $\endgroup$ – dmckee --- ex-moderator kitten Jan 29 '15 at 0:28
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    $\begingroup$ The participants don't stand up straight but instead lean at quite a shallow angle. By varying the angle of lean the players of different heights can keep the rope level. So I don't think the analysis you cite is correct. The requirement to vary the angle of lean to compensate for height differences may well have some effect, but offhand I'm not sure what that would be. $\endgroup$ – John Rennie Jan 29 '15 at 9:58

For calculating the total force a team can generate, it is useful to consider one team as a single system. Note that the total normal force of a team acting on ground only depends on the total mass of the team and the vertical component of tension force of the rope between the teams. You can't do much to increase the team's mass, however you indeed can generate downforce (and upforce for the other team) by putting the tallest closest to the other team (perhaps also the tallest could try to lift the rope even higher, though I'm not sure it will help in practice. The friction force generated by real people is not purely proportional to the normal force).

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I feel like there are three main factors to this question you are trying to highlight...

  1. The strength of each person
  2. The height variation of each person
  3. The amount each person leans when pulling the rope

Let's look at this as an ideal situation to your proposal. The strength of each individual are the same, the height variation is a steady slope, and the leaning of each person continues this steady slope. The other thing I will do to diminish variables is make these individuals pull a rope attached to a wall.

The X direction is the force of the team pulling in one direction and the force of the wall pulling on the team in the opposite direction. (I am generalizing the equations, there is also tension, but the total force in the X direction is zero because the wall is pulling back with the same force) THIS WILL BE THE SAME NO MATTER WHAT ORDER THE TEAM STANDS IN

The Y direction, however, is where your question gets interesting because one thing I did not account for in the X direction is the force of friction of the shoes on the grounds surface.

Now the force of friction is equal to the normal force times the coefficient of friction. The normal force is equal to the weight of each person, in this particular case (mass times gravitational acceleration).

This means that the height of each person does not affect the force to pull the rope, it actually has to do with the total weight of each person and there individual strength.

Edit: Each person is trying to pull the rope horizontally. So lets say we use your example where the short person is in front. The tall person in the back is trying to pull the rope horizontally, it would be wasted force to try and pull it at any angle other than parallel with the surface.

The fact that the last person is taller will put a slight force in the upwards direction of the shorter person (very insignificant) Now if the tall person was in the front he would have a slight force pulling down (the same force as the short person felt only in the opposite direction)

These two forces are very insignificant compared to the weight of the individuals that their heights would have to be that of a toddler to an full grown adult, and even then the force would still be very small.

So yes the heights would make a difference in an ideal situation but even then it would be very very minimal.

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  • $\begingroup$ I think you missed a crucial point in the question, namely, how do alternative orderings affect the normal force of each member of a team? $\endgroup$ – Nit Jul 1 '15 at 21:12
  • $\begingroup$ They don't, height does not affect the problem in any significant way is what I was trying to say, because the main factors are the weight of the people and their strength. $\endgroup$ – papatrott Jul 1 '15 at 21:16
  • $\begingroup$ I don't see you addressing the height differences and the upwards force from other players in any way in your answer as far as I can tell. $\endgroup$ – Nit Jul 1 '15 at 21:18
  • $\begingroup$ I edited my answer for more clarification, I saw your confusion. Hope this makes it more clear. $\endgroup$ – papatrott Jul 1 '15 at 21:31

The drawing has it completely wrong, because it doesn't take into account the leaning that players do. Leaning is beneficial to the team, because it allows you to use gravitational force as well as your muscles to pull the rope and removes the height differences in the rope that would put shorter players at a greater disadvantage than they already have (taller players lean more, shorter ones lean less). Since taller people need a lot of space behind them to lean properly, it is beneficial to the team to have tall people in the back and short people in the front.

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    $\begingroup$ I doubt this approach is correct. There's a limit to how much you can lean before you slip so all in all taller people will still grip the rope at a greater height, even if at a smaller difference. As for spacing, you can space players out in any order so that point doesn't seem to make sense. $\endgroup$ – Nit Jul 1 '15 at 16:28
  • $\begingroup$ @Nit the length of the rope limits the spacing you can use $\endgroup$ – user2425429 Jul 1 '15 at 16:42
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    $\begingroup$ In settings I've seen tug of war take place (I'm sure there are numerous different rule sets around the world), you're usually not that short on rope. $\endgroup$ – Nit Jul 1 '15 at 16:44

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