Solution to Riddle of the Week: The Great American Rail-Trail
This is a solution to Can You Solve the Great American Rail Trail Riddle?, part of our Riddle of the Week series.
Can You Solve the Great American Rail-Trail Riddle?
This problem was suggested by the physicist P. Jeffrey Ungar.
Finally, the Great American Rail-Trail across the whole country is complete! Go ahead, pat yourself on the back—you’ve just installed the longest handrail in the history of the world, with 4,000 miles from beginning to end. But just after the opening ceremony, your assistant reminds you that the metal you used for the handrail expands slightly in summer, so that its length will increase by one inch in total.
“Ha!” you say, “One inch in a 4,000 mile handrail? That’s nothing!” But … are you right?
Let’s suppose when the handrail expands, it buckles upward at its weakest point, which is in the center. How much higher will pedestrians in the middle of the country have to reach in summer to grab the handrail? That is, in the figure below, what is h? (For the purposes of this question, ignore the curvature of the Earth and assume the trail is a straight line.)
SOLUTION: To solve this problem, first let’s fill out the distances in the picture.
1. Half of 4,000 miles is 2,000 miles, which is about 126.7 million inches.
2. The handrail expands by 1 inch in total, so each of the top sides of the triangle will be 126.7 million and ½ inches.
Now, we can use Pythagorean’s Theorem (a2 + b2 = c2) to solve for h:
127,000,0002 + h2 = 127,000,000.52
Solving for h, we get: h = 11,260 inches, or 938 feet. Let’s just say that pedestrians would have to have extremely long arms to still make use of the handrail in summer.
Ungar gives some context for the real-world implications of this problem:
“It takes a surprisingly large perpendicular displacement to accommodate the extra length, which is a general geometric feature of buckling, and the simple triangle model makes a great illustration. Something like a long rail will be more likely to push in and out sideways over shorter baselines, making for smaller but still impressive displacements, as seen in these real-life buckled railways.
Two ways that engineers deal with situations like these rails are by using sliding joints and by optimizing the temperature where a rail held fixed by its supports is stress-free. Sliding joints work by allowing connected rails to expand and contract, while the second technique reduces temperature-related stresses so the rail does not break off its supports.”
Want to talk about brain teasers or try to out-riddle the riddler? Find Laura on Twitter at @LauraFeiveson.
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Published at Fri, 29 Oct 2021 19:15:00 +0000