# Application of Mechanical Similarity

• msbell1
In summary, the author is having difficulty solving a problem at the end of section 10 in Mechanics by Landau and Lifgarbagez. He is trying both methods and feels like he didn't solve it the right way. He is looking for help from someone who is more experienced in the book.
msbell1
Hi,

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!

"The same path for the particle" means that $$\alpha \equiv 1$$ for coordinates:
$$\frac{l'}{l}=\alpha \equiv 1$$
Having differnt masses, the kinetic energy is multiplied not by $$\alpha^2/\beta^2}$$ (see your book) but $$\gamma \alpha^2 / \beta^2$$, where $$\gamma=m'/m$$ and $$\beta=t'/t$$.
To leave equations of motion unaltered, we must have
$$\gamma \frac{\alpha^2}{\beta^2} = \alpha^k$$
where the left side represents the kinetic energy factor and the right side represents the potential energy factor.
Substituting into this equation $$\alpha=1$$ one can have:
$$\gamma=\beta^2$$ or $$\beta = \sqrt{\gamma}$$
or, finally:
$$t'/t = \sqrt{m'/m}$$

I hope this will help you to solve the second problem in the paragraph also :)

Good luck!

Thank you very much--that helped a lot!

And now I will display my solution to the second question (actually it's very similar to the first one--thanks again, by the way).

$$\frac{l'}{l} = \alpha = 1$$(same path again)

This time the kinetic energy is just changed by $$\frac{\alpha^{2}}{\beta^{2}}$$ where $$\beta = \frac{t'}{t}$$

Since potential energies differ by a constant factor, we can write $$\frac{U'}{U} = \gamma$$

Finally, we have $$\frac{\alpha^{2}}{\beta^{2}} = \gamma\alpha^{k}$$

Since $$\alpha = 1$$, we are left with

$$\frac{1}{\beta^{2}} = \gamma$$

$$\beta = \frac{1}{\sqrt{\gamma}}$$

$$\beta = \frac{1}{\sqrt{U'/U}}$$

$$\beta = \sqrt{\frac{U}{U'}}$$

msbell1, sounds good! keep working on Landau & Lifgarbagez.

## What is mechanical similarity?

Mechanical similarity is a concept in engineering and physics that describes the relation between two systems or objects that behave similarly under the same conditions. This means that their physical properties, such as size, shape, and material, are scaled in the same way, resulting in similar performance or behavior.

## Why is mechanical similarity important?

Mechanical similarity is important because it allows scientists and engineers to study and test complex systems or objects in a simplified way. By using scaled models, we can understand the behavior and performance of a larger system without having to build and test the full-scale version, which can be costly and time-consuming.

## What are the key factors for achieving mechanical similarity?

The key factors for achieving mechanical similarity are geometric similarity, kinematic similarity, and dynamic similarity. Geometric similarity refers to having the same shape and proportions, while kinematic similarity means that the motion or movement of the objects is the same. Dynamic similarity involves having the same forces and accelerations acting on the objects.

## How is mechanical similarity used in real-world applications?

Mechanical similarity is used in a wide range of real-world applications, from designing and testing aircraft to studying the flow of fluids in pipelines. It is also used in developing new products and materials, such as medical implants and prosthetics, where it is essential to ensure that the scaled model behaves similarly to the actual product.

## What are the limitations of mechanical similarity?

While mechanical similarity is a useful concept, it has its limitations. One limitation is that it assumes the objects or systems are made of the same material, which may not always be the case in real-world scenarios. Another limitation is that it does not take into account the effects of temperature, pressure, or other environmental factors, which can influence the behavior of the objects being studied.

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