Two years ago I wrote about how to climb a ladder safely without fear. This fellow has no fear and has done the opposite. This fellow has chosen to put a ladder on a table to reach higher than he could otherwise. That table is on another table. At first things are going pretty well, but somewhere about ten steps up the ladder there is disaster. A ladder that held steadily, slips to the edge of the table, and then the table tips over. It’s just physics: the higher he climbs on the ladder the more the horizontal force. Eventually, the force is enough to move the table. He could have got up safely if he moved the tables closer to the wall or if he moved the ladder bottom further to the right on the top table. Either activity would have decreased the slip force, and thus the tendency for the table to tip.
Perhaps the following analysis will help. Lets assume that the ladder is 12.5′ long and sits against a ten foot ledge, with a base 7.5′ away from the wall. Now lets consider the torque and force balance at the bottom of the ladder. Torque is measured in foot-pounds, that is by the rotational product of force and distance. As the fellow climbs the ladder, his weight moves further to the right. This would increase the tendency for the ladder to rotate, but any rotation tendency is matched by force from the ledge. The force of the ledge gets higher the further up the ladder he goes. Let’s assume the ladder weighs 60 lbs and the fellow weighs 240 pounds. When the fellow has gone up ten feet up, he has moved over to the right by 7.5 feet, as the diagram shows. The weight of the man and the ladder produces a rotation torque on the bottom of 60 x 3.75 + 240 x 7.5 = 1925 foot pounds. This torque is combatted by a force of 1926 foot pounds provided by the ledge. Since the ladder is 12.5 feet long the force of the ledge is 1925/12.5 = 154 pounds, normal to the ladder. The effect of this 154 lbs of normal force is to push the ladder to the left by 123.2 lbs and to lift the ladder by 92.4lbs. It is this 123.2 pounds of sideways push force that will cause the ladder to slip.
The slip resistance at the bottom of the ladder equals the net weight times a coefficient of friction. The net weight here equals 60+240-92.4 = 217.6 lbs. Now lets assume that the coefficient of friction is 0.5. We’d find that the maximum friction force, the force available to stop a slip is 217.6 x 0.5 = 108.8 lbs. This is not equal to the horizontal push to prevent rotation, 123.2 lbs. The net result, depending on how you loot at things, is either that the ladder rotates to the right, or that the ladder slips to the left. It keeps slipping till, somewhere near the end of the table, the table tips over.
I occasionally do this sort of detailed physics; you might as well understand what you see in enough detail to be able to calculate what will happen. One take home from here is that it pays to have a ladder with rubber feet (my ladders do). That adds to the coefficient of friction at the bottom.
Robert Buxbaum, November 6, 2019.