﻿ Page of High School Student - Education - Saratov group of theoretical nonlinear dynamics

# Page of High School Student

In this section we present problems in physics for students of high school. You can send your solutions to Alexey Savin to SavinA@info.sgu.ru, and then your work will be checked and you will receive a mark or a referee report.

## For those who starts to learn physics

Here are the problems for beginners in physics that introduce you into a spirit, stile, and approaches of this science. The first impression that is often emphasized in school lessons is that physics is an exact science, somewhat like application of mathematics to a study of nature. It is true that mathematics is important for physics, but this is in fact a very distinct discipline deeply connected with the Universe, having its own approaches and criteria of effectiveness. We hope that the suggested problems will help you to catch and understand some important and interesting peculiarities of physics.

1. Compare pressures of man and cat onto the Earth surface.
2. In ancient ages some people thought that the Earth is placed on the back of a giant whale swimming in the Ocean. Ectimate characteristic size of the whale. Assume that the Earth has a form of semi-sphere of radius R=6400 km, the mean density of it is , density of the whale . Approximate the whale by a cylinder of diameter 10 times less than its length.
3. What is larger - a ratio of the Sun mass to the mass of Earth, or ratio of mass of the oxygen molecule to the mass of electron?
4. A cylinder vessel of height H with cross-section area S is filled by water up to level h. Then, a body of volume V and density is thrown into the vessel. What are possible qualitatively distinct situations. What are parameter intervals necessary for these situations to arise?

For those who are familiar with the school level physics we present here several problems from physical Olympiads.

1. Estimate relative compression of planet Jupiter near its poles, ( is difference of equatorial and polar radii). The mean radius of the planet is r=70000 km, gravitational acceleration near the surface g = 20 m/s2, period of revolution T is equal to 10 hours. Assume for simplicity that the overwhelming part of the whole mass is concentrated in a compact central nuclear of the planet.
2. There are two springs made of the same material twisted in such way that neighbor coils of each spring touch each other. Diameters of the springs are 3 and 9 mm, lengths 1 and 7 sm, diameters of the wire, respectively, 0.2 and 0.6 мм. Spring coefficient for the first is 15 N/m2. Find the coefficient for the second spring.
3. There is a circuit of connected successively a capacitor C and an inductance L, and a voltage is supplied varying in time according to an expression . Determine current in the circuit.

## Problems of research character

Here we suggest you to try problems of nature different from those you deal at lessons or Olympiads. Lesson or Olympiad are short time, and you can spare not so much time to each problem because need to solve as many of them as possible. Among the following problems you may prefer to select one, that you like more. It may take not an hour and not a day to elaborate a solution. Significant is not number of the solved problem, but the depth of understanding and development of the solution. Such problems help you to understand better science as a profession that consists in obtaining novel results by your own hands.

A feature of these problems is that they do not have one definite solution, each of them allows multiplicity of approaches as well as subsequent development. Arsenal of your tools is not fixed. You may use theoretical reasoning, experiments, computer simulation, following your own preferences, experience, consultations, literature, internet, discussion with friends, any other available ways. Discuss the problem with friends.

If your investigation will be successful, you can present results to a school-conference "Nonlinear days for youth in Saratov". You must account, however, that preparation of the report on your work is itself a serious work even if the results are ready, true, and relevant. This is one more step to the profession of a scientist. The authors of the best solutions will be awarded with special diploma.

1. A distorsion of the ocean surface. P.L.Kapitza suggested such a problem: Upside the ocean surface let us place a massive particle of mass m, at the height h up the unperturbed water surface. Study a form of the surface perturbed by the gravitational field of the particle. Depict the corresponding profiles using computer. Is it possible to observe all found forms in realistic experiments on the Earth? Consider possible modifications of the problem (particle under water, case of two comparable gravitational centers etc.)
2. Optical caustics in a cylinder cup. Inside a cylindrical vessel (say, a cup with milk) illuminated by incident light you can observe a bright curve with yet more bright spike (cusp point). The curve is called caustic, and it represents envelope curve of the family of rays reflected from the cylindrical surface. Undertake experimental, theoretical, and computer investigation of the caustic.
Saratov group
of theoretical nonlinear
dynamics