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msbell1

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I am working on Mechanics by Landau and Lifgarbagez, and I have almost completed section 10 on mechanical similarity (in Volume 1, 3rd edition). However, I'm having some difficulty solving the problems at the end of the section. I have tried both of them, and I even found the correct answer to the first problem, but I feel like I didn't solve it the right way, or that there must be a more general way to solve it. So here's the problem.

Find the ratios of times in the same path for particles having different masses, but the same potential energy.

The answer is t'/t = sqrt(m'/m)

Here's how I solved it. First, by reading the question I thought hmmmmm...this sounds like 2 different masses on springs (the springs are the same) oscillating with the same amplitude. So then I approached the solution with this picture in mind.

For a mass on a spring, I know the period is given by T=2pi/w where w = sqrt(k/m)

so t'/t = T'/T = (2pi/sqrt(k/m'))/(2pi/sqrt(k/m)) = sqrt(m'/k)/sqrt(m/k) = sqrt(m'/m)

Ok, so I found the answer that I was supposed to find, but this surely wasn't the approach that I was supposed to take. The authors have not even covered oscillations yet in this book. Can someone please help me solve this problem the way the authors intended it to be solved? Thanks!