Water is flowing in a pipe with a varying cross-sectional area, and at all points the water completely fills the pipe. At point 1 the cross-sectional area of the pipe is 0.070 m^2, and the magnitude of the fluid velocity is 3.50 m/s. (c) Calculate the volume of water discharged from the open end of the pipe in 1.00 hour.

Verified step by step guidance

1. First, use the continuity equation for incompressible fluids, which states that the product of cross-sectional area (A) and fluid velocity (v) at any two points in a pipe must be constant. This can be expressed as A1 * v1 = A2 * v2, where A1 and v1 are the area and velocity at point 1, and A2 and v2 are the area and velocity at the open end of the pipe.
2. Since the problem states that the water completely fills the pipe at all points, and the area at point 1 is given, use the continuity equation to find the velocity at the open end of the pipe (v2) by rearranging the equation to v2 = (A1 * v1) / A2. Note that A2 is the cross-sectional area at the open end of the pipe.
3. Calculate the volume flow rate (Q) at the open end of the pipe, which is given by the formula Q = A2 * v2. This represents the volume of water flowing per unit time through the open end.
4. To find the total volume of water discharged in 1.00 hour, convert the time from hours to seconds (since the flow rate is typically in cubic meters per second). Multiply the volume flow rate by the total time in seconds.
5. The result from the previous step will give you the total volume of water discharged from the open end of the pipe in cubic meters.

Key Concepts

Continuity Equation

The Continuity Equation states that for an incompressible fluid flowing through a pipe, the product of the cross-sectional area and the fluid velocity remains constant along the flow. This principle implies that if the area decreases, the velocity must increase, and vice versa, ensuring mass conservation in the flow.

Flow Rate

Flow rate is defined as the volume of fluid that passes through a given surface per unit time. It can be calculated using the formula Q = A × v, where Q is the flow rate, A is the cross-sectional area, and v is the fluid velocity. This concept is crucial for determining how much fluid is discharged over a specific time period.

Volume Calculation

To calculate the volume of water discharged over a period, one can multiply the flow rate by the time duration. For example, if the flow rate is known, the total volume discharged in one hour can be found using the formula Volume = Flow Rate × Time, where time is converted to seconds for consistency in units.