IZhO 2022 — Задача 1. Problem 1 (10.0 points)

XVIII International Zhautykov Olympiad/Theoretical Competition · 16 февруари 2022 г. · 10 т.

Условие

This problem consists of three independent parts.

Problem 1.1 A ball of radius $R$ rests on the edge of a horizontal table of height $h = 3R$. The ball starts to fall without a significant initial push, and there is no slipping with the edge during the entire time of motion. Find at what distance $l$ from the base of the table the ball first touches the floor. The free fall acceleration is equal to $g$, air resistance is negligible. It is known that the kinetic energy of a ball rolling without slipping is equal to $E = \frac{7}{10}mv^2$, where $v$ denotes the speed of the ball center with $m$ being its mass. [4 т.]

Problem 1.2 A quasi-static process is carried out with one mole of an ideal monatomic gas, as a result of which its initial volume $V_0 = 1\ m^3$ increases four times, and the initial pressure $p_0 = 10^5\ \mathrm{Pa}$ decreases two times. It is known that for each small section of the quasi-static process, the ratio of work to the change in internal energy is a constant value. Find the total work $A$ done by the gas in this process. [3 т.]

Problem 1.3 A thin-walled dielectric hemisphere, negatively charged with the surface density $-\sigma$, is placed on the horizontal table. A point-like ball of mass $m$ is carefully placed on its top. Determine the minimum positive charge $Q$ of the ball, such that it is still in a state of stable equilibrium at the top of the hemisphere. The charges between the hemisphere and the ball are not redistributed. The free fall acceleration is $g$. [3 т.]

Отговори

Решение

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Оригинал в Архива: IZhO-2022-Theory_eng.docx · официални решения: IZhO-2022-Theory_eng_sol.pdf