Torricelli’s theorem – problems and solutions

Torricelli’s theorem – problems and solutions

1. A tube with the height of 100 cm filled with water. A hole Q located at 10 cm above the ground. What is the horizontal distance (x)?

જાણીતા:

Distance between hole and the surface of the water (h) = 100 cm – 10 cm = 90 cm = 0.9 m

ગુરુત્વાકર્ષણને કારણે પ્રવેગક (g) = 10 મીટર/સેકન્ડ²

ઇચ્છિત: Distance of x

ઉકેલો:

The speed of the water flow at the hole

v= ઝડપ, જી = ગુરુત્વાકર્ષણને કારણે પ્રવેગ, h = distance between the hole and the surface of the water

The speed of the water flow at the hole :

હવામાં સમય

The motion of water from the hole to the ground is the અસ્ત્ર ગતિ. 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, ઊભી ગતિ analyzed as free fall motion.

Calculate time in air using the equation of the મુક્ત પતન ગતિ.

જાણીતા:

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

ગુરુત્વાકર્ષણને કારણે પ્રવેગ (g) = 10 m/s²

જોઈતું હતું: સમય અંતરાલ (t)

ઉકેલો:

y = 1/2 g t²

૨૦૦૦ = ૧/૨ (૧૦) ટ²

0.1 = 5 ટી²

t² = 0.1/5

t² = 0.02

t = √0.02 સેકન્ડ

The horizontal distance (x) :

જાણીતા:

પ્રારંભિક વેગ (v_o = વી_ox) = 3√2 m/s

Time in air (t)= √0.02 seconds

જોઈતું હતું: The horizontal distance (x)

ઉકેલો:

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.

જાણીતા:

Distance between hole and the surface of the water (h) = 100 cm – 80 cm = 20 cm = 0.2 m

ગુરુત્વાકર્ષણને કારણે પ્રવેગ (g) = 10 m/s²

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

ઉકેલો:

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…

જાણીતા:

Height (h) = 85 cm – 40 cm = 45 cm = 0.45 meters

ગુરુત્વાકર્ષણને કારણે પ્રવેગ (g) = 10 m/s²

ઇચ્છિત: Speed of water (v)

ઉકેલો:

Torricelli’s theorem states that the velocity of water through a hole distant h from the surface of water equals the ની ગતિ મુક્ત પતનઆઈએનજી water from a height of h.

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

v_t² = 2 ઘનફળ = 2(10)(0.45) = 9

v_t = √9 = 3 મી/સેકન્ડ

4. A ટબ 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)

જાણીતા:

Height (h) = 1.5 m – 0.25 m = 1.25 meters

ગુરુત્વાકર્ષણને કારણે પ્રવેગ (g) = 10 m/s²

જોઈતું હતું: Speed of water (વી)

ઉકેલો:

v_t² = 2 ઘનફળ = 2(10)(1.25) = 25

v_t = √25 = 5 મી/સેકન્ડ

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 …

જાણીતા:

Height (h) = 1 m – 0.20 m = 0.8 meters

ગુરુત્વાકર્ષણને કારણે પ્રવેગ (g) = 10 m/s²

જોઈતું હતું: Speed of water (વી)

ઉકેલો:

v_t² = 2 ઘનફળ = 2(10)(0.8) = 16

v_t = √16 = 4 મી/સેકન્ડ

1. What does Torricelli’s theorem describe?
   • જવાબ: 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?
   • જવાબ: The theorem is expressed as v=√2gh'જ્યાં $v$ 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?
   • જવાબ: 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?
   • જવાબ: 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?
   • જવાબ: 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?
   • જવાબ: 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?
   • જવાબ: 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?
   • જવાબ: 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?
   • જવાબ: 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?

• જવાબ: 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.