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

1. Identify the relevant physical principles: For part (a), use Torricelli's Law, which states that the speed of efflux, v, of a fluid under gravity from a hole in a tank is given by the equation v = \sqrt{2gh}, where g is the acceleration due to gravity (approximately 9.81 m/s^2) and h is the height of the water column above the hole.
2. Calculate the speed of efflux: Substitute the given height of the water column (14.0 m) into the equation from Torricelli's Law to find the speed of the water as it exits the hole.
3. For part (b), determine the volume flow rate, Q, which is the volume of water discharged per unit time. Use the equation Q = A \cdot v, where A is the cross-sectional area of the hole and v is the speed of efflux calculated in part (a).
4. Calculate the cross-sectional area of the hole: Since the hole is circular with a diameter of 6.00 mm, convert this measurement to meters (0.006 m) and use the area formula for a circle, A = \pi (d/2)^2, to find the area.
5. Substitute the area of the hole and the speed of efflux into the formula for volume flow rate to find the volume of water discharged per second.

Key Concepts

Torricelli's Law

Torricelli's Law states that the speed of efflux of a fluid under the force of gravity through a small opening is given by the equation v = √(2gh), where v is the speed of the fluid, g is the acceleration due to gravity, and h is the height of the fluid above the opening. This principle is essential for calculating the speed of water exiting the hole in the tank.

Continuity Equation

The Continuity Equation in fluid dynamics states that for an incompressible fluid, the product of the cross-sectional area and the velocity of the fluid remains constant along a streamline. This concept helps in determining the volume flow rate of water as it exits the hole, which is crucial for calculating the volume discharged per second.

Bernoulli's Principle

Bernoulli's Principle relates the pressure, velocity, and height of a fluid in steady flow. It implies that an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or potential energy. Understanding this principle is important for analyzing the behavior of water as it flows out of the tank and how pressure differences affect the flow rate.