Sep 08, 2023 07:00:00

A discussion will be held on how to create an optimal circular track without applying any load using Lego blocks.

The block toy brand ``

LEGO '' sells `` DUPLO '', a large-sized series that is easy to handle for children from 1 and a half years old. A father whose child made a track using Duplo rails is discussing the best way to arrange the rails.

mathematics - Is this duplo train track under too much tension? - Puzzling Stack Exchange
https://puzzling.stackexchange.com/questions/122232/is-this-duplo-train-track-under-too-much-tension

Below is an image of the track that Mr.

Lezzup 's children made using Duplo rails. According to Lezzup, this track is constructed with only two types of rails: curved rails and straight rails.

The curved rail that Mr. Lezzup's children used can create a loop-shaped track with 12 pieces. However, the children used 13 curved rails.

When creating a loop-shaped track using 13 curved rails, some of the curved rails must be fitted by force, which places an excessive load on each rail.

It has been pointed out that if a load is applied to the rails and the rails become misaligned, the rails may become deformed or damaged in the worst case scenario.

So Lezzup posted questions on Puzzling , asking, ``Is there a way to quickly tell if a rail is under load?'' ``If I need to remove one curved or straight rail, which one is best?'' I posted a question like, 'If I need to add an extra rail, where is the best place to insert it?'

User 2012rcampion answered Mr. Lezzup's question. Regarding ``Is there a load on the rail?'' Mr. 2012rcamipon says that it can be determined that there is no load on the rail by summing the vectors from the start point to the end point of each rail and checking whether the sum of the vectors is zero. claims. According to Mr. 2012rcampion, the presence or absence of load can be calculated using ' trigonometry '.

Also, in the answer by Weather Vane , a method to check the load on each rail is that when there are 12 curved rails, ``Measure the angle of each straight rail, and if the sum of their sines is 0 and the cosine is If the total is 0, it means there is no load.

As a result of calculations by Mr. 2012rcampion, the formula for calculating the sum of the vectors of the rails in the track is (4.5−3√3−2+0.5√3). When we calculated the vectors of the tracks created by Mr. Lezzup's children, it became clear that the sum was not zero. In other words, the load is on the rail. In addition, Mr. 2012 rcampion shows the shape of the rail when no load is applied, and it is the figure below. If there was no load on the rail, the track loop would be broken.

Furthermore, Mr. 2012rcampion explains the rails that should be added and removed using the image below. According to Mr. 2012 rcampion, the curved rail in the area surrounded by the red frame should be removed and the straight rail should be added in the area indicated by 'Add straight'.

By adding and removing rails appropriately, you can create looped tracks with less load.