Question

На гладкой горизонтально расположенной спице в поле тяжести висит шнурок длиной 0.5 м. От небольшого толчка шнурок начинает скользить по спице, не отрываясь от ее поверхности. Найти скорость шнурка, когда с одной из сторон спицы свешивается 0.2 части длины шнурка. Радиус спицы много меньше длины шнурка. Ускорение свободного падения считать равным 10 м/с^2

Answer 1

Введем нулевой уровень потенциальной энергии на начальных высотах центров масс равных свисающих концов (на расстоянии L/4 вниз от спицы, где L-длина шнурка)

когда с одной из сторон спицы свешивается k=0.2 части длины шнурка центр масс короткого конца находится на высоте

$h_1 = L/4 - kL/2 = (1-2k)L/4$

вверх от нуля потенциальной энергии, а центр масс длинного конца на расстоянии

$h_2 = (1-k)L/2-L/4 = (1-2k)L/4$

вниз от нуля потенциальной энергии (что достаточно очевидно).
Весь шнур двигается с одной и той же скоростью, и его энергия равна массе шнура, умноженной на половину квадрата этой скорости, поэтому
Закон Сохранения Энергии

$0 = kmgh_1-(1-k)mgh_2+mv^2/2\\ v^2/2 = -kg(1-2k)L/4 + (1-k)g(1-2k)L/4 = (1-2k)^2gL/4\\\\ v = (1-2k)\sqrt{gL/2}$

Answer 2

Problem Analysis

We are given a scenario where a string is hanging from a smooth, horizontally placed spoke. The string is given a small push and starts sliding along the spoke without detaching from its surface. We need to find the speed of the string when it hangs 0.2 parts of its length from one side of the spoke. The length of the string is 0.5 m, and the radius of the spoke is much smaller than the length of the string. The acceleration due to gravity is given as 10 m/s^2.

Solution

To solve this problem, we can use the principle of conservation of mechanical energy. The mechanical energy of the system is conserved as the string slides along the spoke without detaching. Initially, the string has only potential energy, and as it slides down, this potential energy is converted into kinetic energy.

Let's denote the length of the string as L = 0.5 m. We are given that the string hangs 0.2 parts of its length from one side of the spoke. Therefore, the distance traveled by the string along the spoke is 0.2L.

To find the speed of the string, we can equate the initial potential energy to the final kinetic energy.

Calculation

The initial potential energy of the string is given by the formula:

Initial Potential Energy = mgh

Where: - m is the mass of the string (which we can assume to be negligible) - g is the acceleration due to gravity - h is the height from which the string falls

In this case, the height h is equal to the distance traveled by the string along the spoke, which is 0.2L.

The final kinetic energy of the string is given by the formula:

Final Kinetic Energy = (1/2)mv^2

Where: - m is the mass of the string (which we can assume to be negligible) - v is the speed of the string

Since the mass of the string is negligible, we can cancel it out from both equations.

Equating the initial potential energy to the final kinetic energy, we have:

mgh = (1/2)mv^2

Simplifying the equation, we get:

gh = (1/2)v^2

Solving for v, we get:

v = sqrt(2gh)

Substituting the given values, we have:

v = sqrt(2 * 10 * 0.2L)

Simplifying further, we get:

v = sqrt(4L) = 2sqrt(L)

Substituting the value of L = 0.5 m, we have:

v = 2sqrt(0.5) = 2 * 0.707 = 1.414 m/s

Therefore, the speed of the string when it hangs 0.2 parts of its length from one side of the spoke is approximately 1.414 m/s.

Answer

The speed of the string when it hangs 0.2 parts of its length from one side of the spoke is approximately 1.414 m/s.