Question

Шарик на нити длиной L отклонили на угол 90о от вертикали и отпустили безначальной скорости. Какой угол с вертикалью будет составлять нить маятника в момент,когда его полное ускорение направлено горизонтально?

Answer 1

Тангенциальное ускорение aτ=gsinA
Нормальное ускорение an=V^2/R
Т.к. V^2=2gRcosA, и т.к. проекции на вертикаль должны скомпенсироваться, 
приходим к уравнению tgA=2, откуда A=63.4 grad. 

Answer 2

Calculation of the Angle of the Pendulum

To determine the angle that the pendulum string will make with the vertical when its total acceleration is directed horizontally, we can use the principles of simple harmonic motion.

Let's assume that the length of the pendulum string is represented by L.

When the pendulum is released from an initial angle of 90 degrees from the vertical and without any initial velocity, it will start oscillating back and forth. At any given moment during its motion, the pendulum can be considered as a combination of two perpendicular components of motion: one along the vertical direction and the other along the horizontal direction.

At the extreme points of its motion, when the pendulum reaches its maximum displacement on either side, the total acceleration is directed horizontally. This means that the vertical component of acceleration is zero at these points.

To find the angle that the pendulum string will make with the vertical at this moment, we can use the relationship between the acceleration and the angle of displacement in simple harmonic motion.

The acceleration of a pendulum is given by the equation:

a = -ω²x

Where: - a is the acceleration, - ω is the angular frequency, and - x is the displacement from the equilibrium position.

In simple harmonic motion, the angular frequency is given by:

ω = √(g/L)

Where: - g is the acceleration due to gravity, and - L is the length of the pendulum string.

At the extreme points of the pendulum's motion, the displacement x is equal to the length of the pendulum string L.

Substituting these values into the equation for acceleration, we have:

a = -g(L/L) = -g

Since the total acceleration is directed horizontally at these points, the vertical component of acceleration is zero. Therefore, the angle that the pendulum string will make with the vertical at this moment is given by:

tan(θ) = a_horizontal / a_vertical = 0 / (-g) = 0

This means that the pendulum string will be parallel to the vertical direction at the moment when its total acceleration is directed horizontally.

Please note that the angle of the pendulum will change continuously as it oscillates, and this calculation only gives the angle at the extreme points of its motion when the total acceleration is directed horizontally.

Conclusion

When the pendulum is released from an initial angle of 90 degrees from the vertical and without any initial velocity, the pendulum string will be parallel to the vertical direction at the moment when its total acceleration is directed horizontally.