Tag Archives: ladder

Ladder on table, safe till it’s not.

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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.

Force balance of man on ladder. Based on this, I will go through the slippage math in gruesome detail.

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.

Physics of no fear, no fall ladders

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I recently achieved a somewhat mastery over my fear of heights while working on the flat roof of our lab building / factory. I decided to fix the flat roof of our hydrogen engineering company, REB Research (with help from employees), and that required me to climb some 20 feet to the roof to do some work myself and inspect the work of others. I was pretty sure we could tar the roof cheaper and better than the companies we’d used in the past, and decided that the roof  should be painted white over the tar or that silvered tar should be used — see why. So far the roof is holding up pretty well (looks good, no leaks) and my summer air-conditioning bills were lowered as well.

Perhaps the main part of overcoming my fear of heights was practice, but another part was understanding the physics of what it takes to climb a tall ladder safely. Once I was sure I knew what to do, I was far less afraid. As Emil Faber famously said, “Knowledge is good.”

me on tall ladder

Me on tall ladder and forces. It helps to use the step above the roof, and to have a ladder that extends 3-4′ feet past roof level

One big thing I learned (and this isn’t physics), was to not look down, especially when you are going down the ladder. It’s best to look at the ladder and make sure your hands and feet are going where they should. The next trick I learned was to use a tall ladder — one that I could angle at 20° and extends 4 feet above the roof, see figure. Those 4 feet gave me something to hold on to, and something to look at while going on and off the ladder. I found I preferred to go to or from the roof from a rung that was either at the level of the roof, or a half-step above (see figure). By contrast, I found it quite scary to step on a ladder rung that was significantly below roof level even when I had an extended ladder. I bought my ladder from Acme Ladder of Capital St. in Oak Park; a fiberglass ladder, light weight and rot-proof.

I preferred to set the ladder level (with the help of a shim if needed) at an angle about 20° to the wall, see figure. At this angle, I felt certain the ladder would not tip over from the wind or my motion, and that it would not slip at the bottom, see calculations below.

if the force of the wall acts at right angles to the ladder (mostly horizontally), the wall force will depend only on the lever angle and the center of mass for me and the ladder. It will be somewhat less than the total weight of me and the ladder times sin 20°. Since sin 20° is 0.342, I’ll say the wall force will be less than 30% of the total weight, about 65lb. The wall force provides some lift to the ladder, 34.2% of the wall force, about 22 lb, or 10% of the total weight. Mostly, the wall provides horizontal force, 65 lb x cos 20°, or about 60 lbs. This is what keeps the ladder from tipping backward if I make a sudden motion, and this is the force that must be restrained by friction from the ladder feet. At a steeper angle the anti-tip force would be less, but the slip tendency would be less too.

The rest of the total weight of me and the ladder, the 90% of the weight that is not supported by the roof, rests on the ground. This is called the “normal force,” the force in the vertical direction from the ground. The friction force, what keeps the ladder from slipping out while I’m on it, is this “normal force” times the ‘friction factor’ of the ground. The bottom of my ladder has rubber pads, suggesting a likely friction factor of .8, and perhaps more. As the normal force will be about 90% of the total weight, the slip-restraining force is calculated to be at least 72% of this weight, more than double the 28% of weight that the wall pushes with. The difference, some 44% of the weight (100 lbs or so) is what keeps the ladder from slipping, even when I get on and off the ladder. I find that I don’t need a person on the ground for physics reasons, but sometimes found it helped to steady my nerves, especially in a strong wind.

Things are not so rosy if you use a near vertical ladder, with <10° to the wall, or a widely inclined one, >40°. The vertical ladder can tip over, and the widely inclined ladder can slip at the bottom, especially if you climb past the top of the roof or if your ladder is on a slippery surface without rubber feet.

Robert E. Buxbaum Nov 20, 2013. For a visit to our lab, see here. For some thoughts on wind force, and comments on Engineering aesthetics. I owe to Th. Roosevelt the manly idea that overcoming fear is a worthy achievement. Here he is riding a moose. Here are some advantages of our hydrogen generators for gas chromatography.