S3E6: Triangular spring oscillator

Описание: In this episode we take on our first physics problem of season 3: find the cyclic frequency of simple harmonic oscillation of a system of point masses and ideal springs that has the shape of an equilateral triangle.

We solve this problem in three, gradually increasing in their level of mathematical sophistication, ways.

First, we go back to middle school, dust off the Newtonian formalism and look at the given problem through the lens of forces that generate the simple harmonic oscillation of the system at hand. In order to comprehend and construct the forces-based solution of this problem nothing beyond the middle school knowledge of Euclidean plane geometry is needed.

Second, we capitalize on the fact that the law of conservation of total mechanical energy in our system is applicable. From the fact that that total mechanical energy, E = T + U, remains constant over time, it follows that the first time derivative of E must vanish, as in (dE/dt)=0. By differentiating both sides of the last equation, we deduce the answer to this problem. Which, of course, agrees with the result obtained via the analysis of the relevant forces.

Third, for the most sophisticated approach we switch to the Lagrangian formalism, write down the Lagrangian L = T - U of our system, crank that Lagrangian through a series of traditional steps of computing the various derivatives of L and obtain the respective equation of motion of the oscillator at hand, from which we, again, deduce one and the same answer that agrees with the previously obtained result.

We, thus, see that in physics, just like in mathematics the habit of generating as many distinct solutions of one and the same problem as possible should be an integral part of the fabric of studying and doing physics.