Теорема Торрічеллі – задачі та рішення

Теорема Торрічеллі – задачі та рішення

1. Трубка висотою 100 см заповнена водою. Отвір Q розташований на висоті 10 см над землею. Яка горизонтальна відстань (x)?

Відомо:

Відстань між отвором та поверхнею води (h) = 100 см – 10 см = 90 см = 0.9 м

Прискорення за рахунок сили тяжіння (g) = 10 м/с²

Бажано: Distance of x

Рішення:

The speed of the water flow at the hole

v= швидкість, g = прискорення за рахунок сили тяжіння, h = distance between the hole and the surface of the water

The speed of the water flow at the hole :

Час у повітрі

The motion of water from the hole to the ground is the рух снаряда. The projectile motion could be understood by analyzing the horizontal and vertical component of the motion separately. The x motion occurs at a constant velocity and the y motion occurs at a constant acceleration of gravity.

In this problem, вертикальний рух analyzed as free fall motion.

Calculate time in air using the equation of the рух вільного падіння.

Відомо:

The height of hole (y) = 10 cm = 0.1 m

Прискорення вільного падіння (g) = 10 м/с²

Розшукується: Часовий інтервал (t)

Рішення:

y = 1/2 g t²

0.1 = 1/2 (10) т²

0.1 = 5 т²

t² = 0.1/5

t² = 0.02

t = √0.02 seconds

Горизонтальна відстань (x):

Відомо:

The initial velocity (v_o = v_ox) = 3√2 m/s

Time in air (t)= √0.02 seconds

Розшукується: The horizontal distance (x)

Рішення:

v = x / t

x = v t = (3√2)(√0.02) = (3)(1.41)(0.14) = 0.59 = 0.6 meters

2. A tank containing water with height of 1 meter. At point P, there is a hole. What is the speed of water flow at the hole. Acceleration due to gravity is 10 m/s².

Відомо:

Відстань між отвором та поверхнею води (h) = 100 см – 80 см = 20 см = 0.2 м

Прискорення вільного падіння (g) = 10 м/с²

Бажано: Speed of the water flow at the hole (v)

Рішення:

The speed of the water flow at the hole :

3. A large tub contains water and there is a faucet as shown in the picture below. If g = 10 ms^-2, then the water velocity out of the faucet is…

Відомо:

Висота (h) = 85 см – 40 см = 45 см = 0.45 метра

Прискорення вільного падіння (g) = 10 м/с²

Бажано: Speed of water (v)

Рішення:

Torricelli’s theorem states that the velocity of water through a hole distant h from the surface of water equals the speed of вільне падінняІНГ water from a height of h.

Water velocity is calculated using the free fall motion formula v_t² = 2 г/г

v_t² = 2 г/г = 2(10)(0.45) = 9

v_t = √9 = 3 м/с

4. A ванна filled with water and on a wall there is a hole (see figure below). The speed of water coming out of the hole is… (g = 10 ms^-2)

Відомо:

Висота (h) = 1.5 м – 0.25 м = 1.25 метра

Прискорення вільного падіння (g) = 10 м/с²

Розшукується: Speed of water (V)

Рішення:

v_t² = 2 г/г = 2(10)(1.25) = 25

v_t = √25 = 5 м/с

5. A tank containing water as high as 1 meter (g = 10 ms^-2) and on the wall there is a leak hole (see figure below). The speed of water coming out of the hole is …

Відомо:

Висота (h) = 1 м – 0.20 м = 0.8 метра

Прискорення вільного падіння (g) = 10 м/с²

Розшукується: Speed of water (V)

Рішення:

v_t² = 2 г/г = 2(10)(0.8) = 16

v_t = √16 = 4 м/с

1. What does Torricelli’s theorem describe?
   • Відповідь: Torricelli’s theorem relates the speed of fluid flowing out of an orifice to the height of the fluid column above the opening, assuming steady, inviscid (no viscosity), and incompressible flow.
2. How is Torricelli’s theorem mathematically expressed?
   • Відповідь: The theorem is expressed as v=√2gh​, Де $v$ is the speed of the efflux, $g$ is the acceleration due to gravity, and $h$ is the height of the fluid column above the orifice.
3. Under what assumptions was Torricelli’s theorem derived?
   • Відповідь: The theorem assumes that the fluid is incompressible and non-viscous, the flow is steady, and there is no additional energy added to or taken from the fluid.
4. If a container has two holes at different depths, how will the speeds of the fluids emerging from the holes compare?
   • Відповідь: The fluid emerging from the hole closer to the base will have a greater speed than the fluid from the higher hole. This is because the pressure (and thus the potential energy) is greater at deeper depths.
5. Why does the speed of efflux not depend on the shape or cross-sectional area of the container?
   • Відповідь: Torricelli’s theorem only considers the potential energy due to the height of the fluid column above the orifice. The shape of the container doesn’t change this height, so the speed of efflux remains the same.
6. How does the actual speed of the fluid flowing out of an orifice differ from the prediction made by Torricelli’s theorem in real-world situations?
   • Відповідь: In real-world situations, factors like fluid viscosity, turbulence, and the shape of the orifice can affect the actual speed, often making it less than what Torricelli’s theorem predicts.
7. What is the relationship between Torricelli’s theorem and the conservation of energy?
   • Відповідь: Torricelli’s theorem is derived from the conservation of mechanical energy. It equates the potential energy at the fluid’s surface to the kinetic energy at the orifice.
8. If an orifice is present at the very top of a fluid-filled container, how does Torricelli’s theorem describe the efflux speed?
   • Відповідь: The height $h$ above the orifice would be zero, so according to Torricelli’s theorem, the efflux speed would be zero.
9. How does the presence of atmospheric pressure impact the predictions of Torricelli’s theorem?
   • Відповідь: Torricelli’s theorem assumes the container is open to the atmosphere, and thus, atmospheric pressure acts equally across the fluid’s surface. This pressure is cancelled out when considering the pressure difference across the height of the fluid, so the theorem remains valid.
10. What happens to the speed of efflux as the fluid in the container decreases?

• Відповідь: As the fluid level decreases, the height $h$ above the orifice decreases. According to Torricelli’s theorem, the efflux speed would decrease as v=√2gh​.

These questions and answers explore the foundation, implications, and applications of Torricelli’s theorem in fluid dynamics.