Question

Математический маятник длиной l и массой m отклонили на угол φ0 от положения равновесия и сообщили ему начальную скорость , направленную перпендикулярно к нити вверх. Найти силу натяжения маятника в зависимости от угла φ нити с вертикалью. Очень срочно надо....

Answer 1

Пусть начальная скорость грузика равна v0. запишем закон сохранения энергии

mgL (1 - cosφ0) + (m v0²)/2 = mgL (1 - cosφ) + (m u²)/2,

откуда

u² = v0² - 2gL (cosφ0 - cosφ)

напишем уравнение динамики, проецируя силы на ось, направленную вверх вдоль нити

T - mg cosφ = m (u²/L)

с учетом выражения для скорости, получаем

T = m (g cosφ + (v0²/L) - 2g (cosφ0 - cosφ))

T = mg (3 cosφ - 2 cosφ0 + (v0²/(2gL)))

Answer 2

Mathematical Pendulum and Tension Force

A mathematical pendulum with a length l and a mass m is deflected at an angle φ0 from the equilibrium position and given an initial velocity perpendicular to the string upwards. The task is to find the tension force of the pendulum as a function of the angle φ with the vertical.

The tension force T acting on the pendulum can be expressed as a function of the angle φ using the following formula:

T(φ) = mgcos(φ) + mlφ'²

Where: - T(φ) is the tension force as a function of the angle φ - m is the mass of the pendulum - g is the acceleration due to gravity - l is the length of the pendulum - φ is the angle with the vertical - φ' is the angular velocity of the pendulum

The first term mgcos(φ) represents the component of the gravitational force acting along the string, and the second term mlφ'² represents the centrifugal force due to the angular velocity of the pendulum.

This formula accounts for both the gravitational force and the centrifugal force acting on the pendulum at an angle φ from the vertical.

The tension force T acting on the pendulum is dependent on the angle φ and is influenced by both the gravitational and centrifugal forces.