Tag Archives: roof

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.

Paint your factory roof white

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Standing on the flat roof of my lab / factory building, I notice that virtually all of my neighbors’ roofs are black, covered by tar or bitumen. My roof was black too until three weeks ago; the roof was too hot to touch when I’d gone up to patch a leak. That’s not quite egg-frying hot, but I came to believe my repair would last longer if the roof stayed cooler. So, after sealing the leak with tar and bitumen, we added an aluminized over-layer from Ace hardware. The roof is cooler now than before, and I notice a major drop in air conditioner load and use.

My analysis of our roof coating follows; it’s for Detroit, but you can modify it for your location. Sunlight hits the earth carrying 1300 W/m2. Some 300W/m2 scatters as blue light (for why so much scatters, and why the sky is blue, see here). The rest, 1000 W/m2 or 308 Btu/ft2hr, comes through or reflects off clouds on a cloudy day and hits buildings at an angle determined by latitude, time of day, and season of the year.

Detroit is at 42° North latitude so my roof shows an angle of 42° to the sun at noon in mid spring. In summer, the angle is 20°, and in winter about 63°. The sun sinks lower on the horizon through the day, e.g. at two hours before or after noon in mid spring the angle is 51°. On a clear day, with a perfectly black roof, the heating is 308 Btu/ft2hr times the cosine of the angle.

To calculate our average roof heating, I integrated this heat over the full day’s angles using Euler’s method, and included the scatter from clouds plus an absorption factor for the blackness of the roof. The figure below shows the cloud cover for Detroit.

Average cloud cover for Detroit, month by month.

Average cloud cover for Detroit, month by month; the black line is the median cloud cover. On January 1, it is strongly overcast 60% of the time, and hardly ever clear; the median is about 98%. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States

Based on this and an assumed light absorption factor of σ = .9 for tar and σ = .2 after aluminum. I calculate an average of 105 Btu/ft2hr heating during the summer for the original black roof, and 23 Btu/ft2hr after aluminizing. Our roof is still warm, but it’s no longer hot. While most of the absorbed heat leaves the roof by black body radiation or convection, enough enters my lab through 6″ of insulation to cause me to use a lot of air conditioning. I calculate the heat entering this way from the roof temperature. In the summer, an aluminum coat is a clear winner.

Detroit High and Low Temperatures Over the ear

High and Low Temperatures For Detroit, Month by Month. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States

Detroit has a cold winter too, and these are months where I’d benefit from solar heat. I find it’s so cloudy in winter that, even with a black roof, I got less than 5 Btu/ft2hr. Aluminizing reduced this heat to 1.2 Btu/ft2hr, but it also reduces the black-body radiation leaving at night. I should find that I use less heat in winter, but perhaps more in late spring and early fall. I won’t know the details till next year, but that’s the calculation.

The REB Research laboratory is located at 12851 Capital St., Oak Park, MI 48237. We specialize in hydrogen separations and membrane reactors. By Dr. Robert Buxbaum, June 16, 2013