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Здравствуйте! Прошу помощи в следующем вопросе: Куда в данном случае направлены векторные величины момент силы и угловое ускорение?
Свободно вращающаяся рамка с током I = 0.5 А помещается в однородное магнитное поле. Определить индукцию В поля, если период малых колебаний рамки Т = 1 с, площадь рамки S = 3.14[$215$]10–2 м2, Момент инерции J = 10–4 кг[$215$]м2. Ось вращения перпендикулярна вектору магнитной индукции
Условие: Свободно вращающаяся рамка с током I = 0.5 А помещается в однородное магнитное поле. Определить индукцию В поля, если период малых колебаний рамки Т = 1 с, площадь рамки S = 3.14×10–2 м2, Момент инерции J = 10–4 кг×м2. Ось вращения перпендикулярна вектору магнитной индукции. Рисунок в приложении.
Здравствуйте, UnaReider! Начнём с того, что у Вас решение не правильное. Положение равновесия достигается, когда рамка перпендикулярна линиям поля, так что внутри контура магнитное поле проходящего в нём тока совпадает по направлению с внешним магнитным полем. При отклонении [$945$] от положения равновесия момент силы M=-IBS[$183$]sin[$945$] угловое ускорение [$949$]=M/J=-IBS/J[$183$]sin[$945$] Для гармонических колебаний [$949$]=d2[$945$]/dt2=-[$969$]2[$945$] Принимая для малых колебаний sin[$945$]/[$945$][$8776$]1 [$969$]2=-[$949$]/[$945$]=IBS/J[$183$]sin[$945$]/[$945$][$8776$]IBS/J Откуда индукция B=[$969$]2J/IS=(2[$960$]/T)2J/IS[$8776$]0,25 Тл
Учтите также, что в этих формулах [$969$] - угловая частота колебаний, а не угловая скорость вращения рамки. Видимо, в этом месте у Вас и возникает путаница.
А псевдовекторы момента силы и углового ускорения направлены вдоль оси вращения. Если для находящегося сверху наблюдателя вращение ускоряется по часовой стрелке, то эти псевдовекторы направлены вниз.
В принципе, здесь имеет место дифференциальное уравнение [$949$]=d2[$945$]/dt2=M/J=-IBS/J[$183$]sin[$945$] В таком виде оно точного аналитического решения не имеет, но в приближении для малых колебаний получаем [$949$]=d2[$945$]/dt2=-IBS/J[$183$][$945$] Подобным свойством обладают тригонометрические функции. Так, если искать уравнение колебаний в виде [$945$]=A[$183$]sin([$969$]t) то вторая производная [$949$]=d2[$945$]/dt2=-[$969$]2A[$183$]sin([$969$]t)=-[$969$]2[$945$] откуда [$969$]2=-[$949$]/[$945$][$8776$]IBS/J Собственно, для любого вида гармонических колебаний необходимо идентифицировать такую взаимосвязь со второй производной от смещения, чтобы найти выражение угловой частоты.
Попробуйте мысленно покрутить изображение. Так, если слева движение от нас, а справа - к нам, то для наблюдателя верху это вращение по часовой стрелке, поэтому псевдовектор, описывающий это вращение, направлен вниз.
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