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Консультация № 161715
01.03.2009, 17:24
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Найти амплитуду и начальную фазу гармонического колебания, полученного от сложения одинаково направленных колебаний, заданных уравнениями x1 = 0,02sin(5пt + п/2) м и x2 = 0,03sin(5пt + п/4) м
Обсуждение
Неизвестный
04.03.2009, 21:12
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это ответ
Здравствуйте, Нечаева Дарья Алексеевна!
Чтобы сложить колебания воспользуемся следующими формулами: sin(a+b)=sin(a)cos(b)+cos(a)sin(b) (1) и a*sin(x)+b*cos(x)=sqrt(a^2+b^2)*sin(x+arcsin(b/sqrt(a^2+b^2))) (2) (я обозначил корень квадратный - sqrt). Тогда результирующее колебание x=x1+x2=(1)=0,02cos(5пt)+0,03*sqrt(2)(sin(5пt)+cos(5пt))/2=0,03*sqrt(2)*sin(5пt)/2+(0,02+0,03*sqrt(2)/2)*cos(5пt). Обозначим a=0,03*sqrt(2)/2, b=(0,02+0,03*sqrt(2)/2) и воспользуемся формулой (2): x=sqrt(a^2+b^2)*sin(5пt+arcsin(b/sqrt(a^2+b^2))). Формулу конечного колебания получили, теперь осталось подставить числа и найти искомые величины. Амплитуда: А=sqrt(a^2+b^2)=0,046. Начальная фаза [$966$]=arcsin(b/sqrt(a^2+b^2))=63[$186$].
Чтобы сложить колебания воспользуемся следующими формулами: sin(a+b)=sin(a)cos(b)+cos(a)sin(b) (1) и a*sin(x)+b*cos(x)=sqrt(a^2+b^2)*sin(x+arcsin(b/sqrt(a^2+b^2))) (2) (я обозначил корень квадратный - sqrt). Тогда результирующее колебание x=x1+x2=(1)=0,02cos(5пt)+0,03*sqrt(2)(sin(5пt)+cos(5пt))/2=0,03*sqrt(2)*sin(5пt)/2+(0,02+0,03*sqrt(2)/2)*cos(5пt). Обозначим a=0,03*sqrt(2)/2, b=(0,02+0,03*sqrt(2)/2) и воспользуемся формулой (2): x=sqrt(a^2+b^2)*sin(5пt+arcsin(b/sqrt(a^2+b^2))). Формулу конечного колебания получили, теперь осталось подставить числа и найти искомые величины. Амплитуда: А=sqrt(a^2+b^2)=0,046. Начальная фаза [$966$]=arcsin(b/sqrt(a^2+b^2))=63[$186$].
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