Ch 12: Fluid Mechanics

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 12: Fluid Mechanics

Problem 44

Chapter 12, Problem 44

A small circular hole 6.00 mm in diameter is cut in the side of a large water tank, 14.0 m below the water level in the tank. The top of the tank is open to the air. Find (a) the speed of efflux of the water and (b) the volume discharged per second.

Verified step by step guidance

To find the speed of efflux of the water, we can use Torricelli's theorem, which states that the speed \( v \) of efflux of a fluid under gravity through a hole is given by \( v = \sqrt{2gh} \), where \( g \) is the acceleration due to gravity (approximately 9.81 m/s²) and \( h \) is the height of the fluid above the hole. Here, \( h = 14.0 \) m.

Substitute the values into the equation: \( v = \sqrt{2 \times 9.81 \times 14.0} \). This will give you the speed of efflux in meters per second.

Next, to find the volume discharged per second, we need to calculate the area of the hole. The diameter of the hole is 6.00 mm, which is 0.006 m. The area \( A \) of the hole is given by the formula for the area of a circle: \( A = \pi r^2 \), where \( r \) is the radius of the hole.

Calculate the radius: \( r = \frac{0.006}{2} \) m. Then, substitute into the area formula: \( A = \pi \times (0.003)^2 \).

The volume discharged per second, also known as the flow rate \( Q \), is given by \( Q = A \times v \). Use the area calculated in the previous step and the speed of efflux from step 2 to find \( Q \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Torricelli's Law

Torricelli's Law states that the speed of efflux of a fluid under gravity from an orifice is given by v = √(2gh), where v is the speed, g is the acceleration due to gravity, and h is the height of the fluid above the opening. This principle is crucial for determining the speed at which water exits the tank.

Continuity Equation

The Continuity Equation in fluid dynamics expresses the conservation of mass in a flowing fluid. It states that the product of the cross-sectional area and the velocity of the fluid remains constant along a streamline. This concept helps calculate the volume of water discharged per second by relating the speed of efflux to the area of the hole.

Hydrostatic Pressure

Hydrostatic pressure is the pressure exerted by a fluid at equilibrium due to the force of gravity. It increases with depth and is given by P = ρgh, where ρ is the fluid density, g is gravity, and h is the height of the fluid column. Understanding this concept is essential for analyzing the forces driving the water out of the tank.

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