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Консультация № 161177
23.02.2009, 11:40
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Уважаемые профессоры! Помогите с решением задачи.
Найти круговую частоту и амплитуду гармонических колебаний частицы, если на расстояниях Х1 и Х2 от положения равновесия её скорость равна соответственно v1 v2.
Найти круговую частоту и амплитуду гармонических колебаний частицы, если на расстояниях Х1 и Х2 от положения равновесия её скорость равна соответственно v1 v2.
Обсуждение
24.02.2009, 08:39
общий
это ответ
Здравствуйте, Nike0!
Расстояние частицы от положения равновесия определяется уравнением: Х = A*SIN(ω*t), где A - амплитуда, ω - круговая частота, t - текущее время. Скорость частицы V определяется как производная от расстояния по времени: V = dХ/dt = ω*A*COS(ω*t). Для данной задачи имеем: Х1 = A*SIN(ω*t1) (1); V1 = ω*A*COS(ω*t1) (1а). Возведём обе части (1а) в квадрат, а обе части (1) ещё и умножим на ω2: ω2*Х12 = ω2*A2*(SIN(ω*t1))2 (2); V12 = ω2*A2*(COS(ω*t1))2 (2а). Сложив (2) и (2а), получаем: ω2*Х12 + V12 = ω2*A2 (3). Аналогично: ω2*Х22 + V22 = ω2*A2 (4), а вычитая (4) из (3) и решая относительно ω2: ω2 = (V22 - V12)/(Х12 - Х22) (5), а подставив (5) в (3) и решая относительно A2, после упрощений: A2 = ((Х1*V2)2 - (Х2*V1)2)/(V22 - V12) (6). Окончательно: ω = √((V22 - V12)/(Х12 - Х22)) (7); A = √(((Х1*V2)2 - (Х2*V1)2)/(V22 - V12)) (8).
Расстояние частицы от положения равновесия определяется уравнением: Х = A*SIN(ω*t), где A - амплитуда, ω - круговая частота, t - текущее время. Скорость частицы V определяется как производная от расстояния по времени: V = dХ/dt = ω*A*COS(ω*t). Для данной задачи имеем: Х1 = A*SIN(ω*t1) (1); V1 = ω*A*COS(ω*t1) (1а). Возведём обе части (1а) в квадрат, а обе части (1) ещё и умножим на ω2: ω2*Х12 = ω2*A2*(SIN(ω*t1))2 (2); V12 = ω2*A2*(COS(ω*t1))2 (2а). Сложив (2) и (2а), получаем: ω2*Х12 + V12 = ω2*A2 (3). Аналогично: ω2*Х22 + V22 = ω2*A2 (4), а вычитая (4) из (3) и решая относительно ω2: ω2 = (V22 - V12)/(Х12 - Х22) (5), а подставив (5) в (3) и решая относительно A2, после упрощений: A2 = ((Х1*V2)2 - (Х2*V1)2)/(V22 - V12) (6). Окончательно: ω = √((V22 - V12)/(Х12 - Х22)) (7); A = √(((Х1*V2)2 - (Х2*V1)2)/(V22 - V12)) (8).
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