Question

Тело массой 2 кг подвешен на нерастяжимой нити длиной 1 м , отклонено от положения равновесия на угол 20" и отпущено. определите максимальную скорость колеблющегося тела

Answer 1

Воспользуемся законом сохранения энергии. У отклоненного тела есть только потенциальная энергия.

$E = mgh = mgL(1-\cos\alpha)$

Переход от h к углу сделан из геометрических соображений

В положении равновесия вся энергия переходит в кинетическую, поэтому

$\displaystyle mv^2/2 = mgL(1-\cos\alpha)\\\\ v = \sqrt{2gL(1-\cos\alpha)}$

Answer 2

Calculation of the Maximum Velocity of the Oscillating Body

To determine the maximum velocity of the oscillating body, we can use the principle of conservation of mechanical energy. When the body is at its maximum displacement from the equilibrium position, all of its potential energy is converted into kinetic energy. At this point, the body has its maximum velocity.

The potential energy of the body is given by the formula:

Potential energy (PE) = mass (m) * acceleration due to gravity (g) * height (h)

In this case, the height (h) is equal to the length of the string (1 m) multiplied by the sine of the angle of displacement (20°). The acceleration due to gravity (g) is approximately 9.8 m/s^2.

Using these values, we can calculate the potential energy:

PE = 2 kg * 9.8 m/s^2 * 1 m * sin(20°)

Now, we can calculate the maximum velocity (v) using the formula for kinetic energy:

Kinetic energy (KE) = 0.5 * mass (m) * velocity^2 (v^2)

Since the potential energy is converted entirely into kinetic energy at the maximum displacement, we can equate the two:

PE = KE

Substituting the values, we have:

2 kg * 9.8 m/s^2 * 1 m * sin(20°) = 0.5 * 2 kg * v^2

Simplifying the equation, we can solve for the maximum velocity (v):

v^2 = (2 * 9.8 m/s^2 * 1 m * sin(20°)) / 0.5

v^2 = 2 * 9.8 m/s^2 * 1 m * sin(20°)

v = sqrt(2 * 9.8 m/s^2 * 1 m * sin(20°))

Now, we can calculate the maximum velocity using the given values:

v ≈ sqrt(2 * 9.8 m/s^2 * 1 m * sin(20°)) ≈ 2.78 m/s

Therefore, the maximum velocity of the oscillating body is approximately 2.78 m/s.