Question

Гидравлический пресс изготовлен с использованием двух вертикальных цилиндрических сообщающихся сосудов, заполненных жидкостью и закрытых лёгкими поршнями. Если на малый поршень этого пресса положить груз массой 40 кг, удерживая больший поршень неподвижным, то сила давления жидкости на больший поршень будет равна по модулю 900 Н. Во сколько раз радиус большего поршня пресса превосходит радиус меньшего поршня? Подробное решение, пожалуйста

Answer 1

1)
Сила, действующая на малый поршень:
F₁ = m₁*g = 40*10 = 400 Н
Площадь малого поршня:
S₁ = π*(R₁)²

2)
Сила, действующая на большой поршень:
F₂ = 900 H - по условию задачи
Площадь большого  поршня:
S₁ = π*(R₂)²

3)
Запишем закон гидравлического пресса:
S₂/S₁ = F₂/F₁
(R₂/R₁)² = 900 / 400
R₂/R₁ = 30/20 = 3/2    или  радиус большого поршня в полтора раза больше радиуса малого поршня.

Answer 2

Hydraulic Press with Two Communicating Cylindrical Vessels

A hydraulic press is a device that utilizes the principle of Pascal's law to generate a large force using a small force. It consists of two communicating cylindrical vessels filled with a liquid (usually oil) and closed off with lightweight pistons. When a force is applied to the small piston, it is transmitted through the liquid to the larger piston, resulting in a larger force being exerted.

In this case, a hydraulic press is constructed with two vertical cylindrical vessels that communicate with each other. The vessels are filled with a liquid and closed off with lightweight pistons. A load of 40 kg is placed on the small piston, while the larger piston remains stationary. The question asks for the ratio of the radii of the larger piston to the smaller piston.

Solution

To solve this problem, we can apply Pascal's law, which states that the pressure exerted by a fluid is transmitted equally in all directions. Therefore, the pressure exerted by the liquid on the small piston is equal to the pressure exerted on the large piston.

Let's denote the radius of the smaller piston as r1 and the radius of the larger piston as r2. We are given that the force exerted by the liquid on the larger piston is 900 N and the load on the smaller piston is 40 kg.

To find the ratio of the radii, we can use the formula for pressure:

Pressure = Force / Area

The area of a piston can be calculated using the formula for the area of a circle:

Area = π * radius^2

Since the pressure is the same on both pistons, we can set up the following equation:

Pressure on small piston = Pressure on large piston

Force on small piston / Area of small piston = Force on large piston / Area of large piston

Substituting the formulas for force and area, we get:

(40 kg * 9.8 m/s^2) / (π * r1^2) = 900 N / (π * r2^2)

Simplifying the equation, we can cancel out the common factors:

40 * 9.8 / r1^2 = 900 / r2^2

Now, we can solve for the ratio of the radii:

(r2^2) / (r1^2) = (40 * 9.8) / 900

Taking the square root of both sides, we get:

r2 / r1 = sqrt((40 * 9.8) / 900)

Calculating the value, we find:

r2 / r1 ≈ 1.4

Therefore, the radius of the larger piston is approximately 1.4 times the radius of the smaller piston.

Conclusion

In summary, the ratio of the radius of the larger piston to the radius of the smaller piston in the hydraulic press is approximately 1.4. This is determined by applying Pascal's law and setting the pressure exerted by the liquid on both pistons equal to each other.