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If a man pushes an immovable, rigid wall while standing on rough ground, would the man be in equilibrium?

Here's my attempt:

The man would be in equilibrium in the vertical direction, since gravitational force (due to Earth) on the man; and the normal reaction force from the ground to the man cancel out each other.

The horizontal direction is what I'm not too sure about. The wall would certainly push back on the man. Would friction (due to rough ground) act on the man in the opposite direction? If it does, wouldn't its value be at maximum $\mu N$? Since the man can exert a force larger than it's own weight, shouldn't the equilibrium state depend on the force exerted by the man?

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closed as off-topic by sammy gerbil, Bill N, Jon Custer, Daniel Griscom, Kyle Kanos Apr 2 '18 at 10:16

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  • $\begingroup$ Friction and action of man balances the reaction... $\endgroup$ – Nehal Samee Mar 27 '18 at 14:50
  • $\begingroup$ But what if the forced exerted by the man on the wall is greater than the maximum possible value of static/kinetic friction? $\endgroup$ – skb Mar 27 '18 at 14:52
  • $\begingroup$ Since the wall is immovable, there is no maximum possible value of static/kinetic friction. $\endgroup$ – probably_someone Mar 27 '18 at 14:57
  • $\begingroup$ When we are considering the free body diagram of the man, shouldn't the maximum value of friction acting on him be $\mu N$? Why would the wall influence the friction between the man and the ground? $\endgroup$ – skb Mar 27 '18 at 14:59
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    $\begingroup$ Ah, ok. I thought you were referring to the maximum amount of friction on the bottom of the wall. There is, of course, a maximum possible value of static friction on the man's feet. This is equivalent, by Newton's Third Law, to the maximum amount of force that the man can apply to the wall. $\endgroup$ – probably_someone Mar 27 '18 at 15:14
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We could consider two ways a man could push on the wall, one is by using his weight and the other by using his muscles.

To illustrate and, if needed, to analyze the action of the weight, we could just use a log.

enter image description here

Here red arrows represent forces acting on the log, while blue arrows represent forces the log exerts on the wall and the floor.

From this picture, we can see that the weight is balanced by the vertical component of the floor reaction and the wall friction force, which provides additional support for the log.

The horizontal equilibrium is achieved between the friction force on the floor and the horizontal reaction of the wall.

If the log position is close to vertical, its push on the wall is very weak, since there is a very short lever for the weight to develop any significant force. A man would have to lean, at least a little bit, to produce any horizontal push without using muscles.

If the log position is close to horizontal, the total force on the wall is more significant, but its horizontal component is getting close to nothing. Also, we can expect that, due to the reduced pressure on the floor, the friction force will be very low and the log will be likely to slip.

So, we must conclude that the log will exert a maximum horizontal push on the wall when it is at some intermediate angle and that push will not exceed a fraction of its weight.

The action of man's muscles could be simplistically modeled as a compression spring pinned at some angle between the wall and the floor and pushing on both. As shown on the picture below with green arrows, these spring-like forces originate somewhere in the man's core.

enter image description here

Of course we still have to have equilibrium between all horizontal and vertical forces, but the forces are not limited by man's weight. One of the differences here is that, if the man is moderately strong, the net vertical component of of the force he exerts on the wall could be pointing up (not down as was the case with the log). Correspondingly, the wall is pushing down.

As a result, the pressure on the floor could significantly exceed man's weight and therefore the resulting high friction force could balance very strong horizontal reaction of the wall. The assumption here is that floor and wall surfaces are reasonably coarse.

So, again, by using his muscles, a man can not only significantly increase the horizontal push on the wall (in comparison with the weight based mechanism), but, by directing his push upwards, he can also increase the pressure on the floor and, therefore, the friction force, which can balance the horizontal reaction of the wall.

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