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Задать вопрос экспертам

Консультация № 201042

02.06.2021, 10:33

0.00 руб.

0 1 1

Образование

Здравствуйте! У меня возникли сложности с таким вопросом:
"Материальная точка движется вдоль оси x так, что ее скорость зависит от координаты x по закону v=[(A-B[x][2])][1/2], где А=136[м][2]/[с][2], В=100[с][2]. Показать, что уравнение движения точки является динамическим уравнением гармонических колебаний и найти период Т этих колебаний."

Обсуждение

давно

Студент
405049
133

02.06.2021, 16:10

общий

это ответ

Уравнение гармонических колебаний обычно пишут в виде

d²x/dt² + [$969$]²[$183$]x = 0 или d²x/dt² = - [$969$]²[$183$]x

Поэтому сначала найдем вторую производную, т.е. производную от скорости.

d²x/dt² = d ( (A - B[$183$]x²)^1/2 )/dt= 1/2[$183$](-2[$183$]B[$183$]x)/(A - B[$183$]x²)^1/2)[$183$]dx/dt = -B[$183$]x[$183$]dx/dt/(A - B[$183$]x²)^1/2 = -B[$183$]x[$183$](A - B[$183$]x²)^1/2/(A - B[$183$]x²)^1/2 = -B[$183$]x

то есть d²x/dt² = -B[$183$]x или d²x/dt² + B[$183$]x = 0

Таким образом, мы получили уравнение для гармонических колебаний, где [$969$]²=B.

T=2[$183$][$960$]/[$969$] = 2[$183$][$960$]/sqrt(B) = 2[$183$]3,14/sqrt(100) [$8776$] 0,6 (с)

ОТВЕТ: [$8776$] 0,6 с

5

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