How Do Springs A, B, and C Reach Equilibrium on the 4th and 5th Intervals?

[Werg22]

This one is about bouncing springs. There are three springs bouncing on the floor, A, B and C. Spring A has a period of 2 seconds, spring B a period of 5 seconds and spring C a period of 9 seconds. If they all start at equilibrium position, eventually the springs reach back the equilibrium position on a 3 consecutive second interval (each reaching it once). What will be the orders in which the springs reach the equilibrium position on the 4th and 5th of these intervals?

 

[Werg22]

I'll give the solution here in case you really can't figure it out...

Since B has a period of 5, at the intervals of interest, a number that either ends with 0 or 5 will be included. For the cases in which it ends in 5, we could have

1. x5, x6, x7
2. x4,x5,x6
3. x3,x4,x5

Out of the above, only 1 and 2 respect the condition that each spring has to bounce only once during the interval. So we are looking for a multiple of 9 that either ends in 7 or 3. The first few five are 27, 63, 117, 153, 207.

Now consider the case in for which the second at which spring B reaches back the equilibrium position is a multiple of 10. It gives as possibilities

x0,x1,x2
x8, x9, y0

Out of the above there's no solution that corresponds to the intervals we are looking for. So all the intervals pertain to case 1, and are, in order,

25, 26, 27
63, 64, 65
115, 116, 117
153, 154, 155
205, 206, 207

The order for 4th interval is C - A - B and the order for the 5th interval is B - A - C.

 

[Architeuthis Dux]

I would approach this brain teaser by first identifying the key variables in the scenario. In this case, the variables are the three springs (A, B, and C) and their respective periods (2 seconds, 5 seconds, and 9 seconds). The goal is to determine the order in which the springs will reach the equilibrium position on the 4th and 5th intervals.

To solve this problem, I would use the concept of phase difference in simple harmonic motion. The phase difference between two oscillating objects is the fraction of a cycle by which one object leads or lags behind the other. In this case, the three springs have different periods, which means they will have different phase differences with each other.

Based on the given information, we know that on the 3rd consecutive second interval, each spring will reach the equilibrium position once. This means that all three springs will be in phase with each other at that moment. Therefore, we can conclude that on the 4th and 5th intervals, the springs will reach the equilibrium position in the same order as they did on the 3rd interval.

To determine the order, we need to calculate the phase differences between the springs. The phase difference between spring A and B can be calculated by dividing the difference in their periods (5 seconds - 2 seconds = 3 seconds) by the period of spring B (5 seconds). This gives us a phase difference of 3/5 or 0.6. Similarly, the phase difference between spring B and C is 4/9 or 0.44.

Based on these phase differences, we can conclude that on the 4th interval, spring A will reach the equilibrium position first, followed by spring B and then spring C. On the 5th interval, the order will be spring A, spring C, and then spring B.

In conclusion, the order in which the springs will reach the equilibrium position on the 4th and 5th intervals is determined by their respective phase differences. This brain teaser demonstrates the application of simple harmonic motion principles and the concept of phase difference in solving complex problems.