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Уважаемые эксперты! Подскажите, пожалуйста, с решением таких задач (вроде бы достаточно лёгких): 1. Груз малых размеров, прикреплённый лёгкой пружиной жёсткости k = 200 Н/м к неподвижной точке, движется по окружности радиуса R = 20 см с постоянной по модулю скоростью v = 2 м/с, скользя по гладкой горизонтальной поверхности. Радиус окружности в два раза больше длины пружины в нерастянутом состоянии. Какова масса груза? 2. Груз малых размеров массой m = 1 кг, прикреплённый лёгкой пружиной к неподвижной точке, движется по окружности с постоянной по величине скоростью v = 2 м/с, скользя по гладкой горизонтальной поверхности. Радиус окружности в два раза больше длины пружины в нерастянутом состоянии, равной L = 10 см. Какова жёсткость пружины? Заранее благодарю за ответ. Интересует решение, а не конечный результат.
Обсуждение
Неизвестный
11.06.2006, 13:58
общий
Вот так всегда - спросишь, а потом сам догадываешься, как задача решаетсяВпрочем, ответов всё равно жду :-)
Неизвестный
11.06.2006, 19:43
общий
это ответ
Здравствуйте, Ерёмин Андрей! Центростремительное ускорение тела при движении по окружности <b>a=v²/R</b>. Сила инерции <b>F<sub>ин.</sub>=ma=mv²/R</b>. Сила инерции уравновешивается силой реакции пружины, равной <b>F<sub>пр.</sub>=kx</b>, где x - отклонение конца пружины от ненапряжённого состояния, <b>x=R/2</b>. <b>F<sub>пр.</sub>=kR/2</b>. <b>F<sub>ин.</sub>=F<sub>пр.</sub></b>. <b>mv²/R=kR/2</b>. <b>m=kR²/(2v²)=200*0,2²/(2*2²)=1 кг</b>.
Задачи, действительно, не очень сложные, но пройти мимо возможности поупражняться я не смог... Решения находятся во вложении.
С уважением.
Приложение: 1. Рассмотрим случай пространственного движения. Груз находится в равновесии под действием следующих сил: 1) силы тяжести Fт=mg, направленной вертикально вниз; 2) центробежной силы Fц=(mv^2)/R, направленной по нормали к траектории движения груза от оси вращения к грузу; 3) силы упругости пружины Fу=k*∆L=k((R/sin α)-(R/2))=kR(2-sin α)/(2*sin α), направленной вдоль оси пружины к точке подвеса под углом α к вертикали (α=arctg Fц/Fт=arctg (v^2)/gR). В векторной форме условие равновесия имеет вид: Fу+Fц+Fт=0, а в проекции на ось пружины имеем: Fу=Fц*sin α+Fт*cos α, откудаkR(2-sin α)/(2*sin α)=((mv^2)*sin α/R)+mg*cos α,m=(kR^2)*(2-sin α)/(2*((v^2)*(sin α)^2)+gR*sin 2 α).В случае плоского движения α=π/2, m=(kR^2)/(2v^2). 2. Рассуждая так же, как и при решении предыдущей задачи, имеем: Fц=(mv^2)/(2L), Fу=kL(2-sin α)/sin α, α=arctg (v^2)/(2L), kL(2-sin α)/sin α=((mv^2)*sin α/(2L))+mg*cos α, k=m((v^2)*(sin α)^2)+gL*sin 2 α)/((2L^2)*(2-sin α)).В случае плоского движения k=(mv^2)/(2L^2).Численные ответы можно получить после подстановок.
Об авторе:
Facta loquuntur.
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