Q2. A fluid flows through a fixed (i.e., not rotating) curved pipe of uniform diameter 15 cm with a velocity of the fluid particles on the streamline along the center line of the pipe defined by V = 3t m/sec as shown in the figure below. Determine the normal acceleration and the streamline acceleration when t = 10 sec. 0.5 m +
Q2. A fluid flows through a fixed (i.e., not rotating) curved pipe of uniform diameter 15 cm with a velocity of the fluid particles on the streamline along the center line of the pipe defined by V = 3t m/sec as shown in the figure below. Determine the normal acceleration and the streamline acceleration when t = 10 sec. 0.5 m +
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Please show all steps to solve this problem so your thought process is easy to follow. Also please show all formulas that you used. Subject is fluid mechanics
![**Problem Statement:**
A fluid flows through a fixed (i.e., not rotating) curved pipe of uniform diameter 15 cm with a velocity of the fluid particles on the streamline along the center line of the pipe defined by \( V = 3t \) m/sec. Determine the normal acceleration and the streamline acceleration when \( t = 10 \) sec.
**Diagram Explanation:**
The diagram shows a curved pipe with a uniform diameter. The fluid flows from one end to the other along a streamline. The curvature of the pipe has a radius of 0.5 m. The direction of the fluid flow is indicated by arrows.
---
**Instructions on How to Solve:**
To determine the normal and streamline acceleration at \( t = 10 \) seconds, use the following formulas:
1. **Streamline Acceleration (\( a_s \))**:
The streamline acceleration is the derivative of the velocity function with respect to time \( t \).
\[
a_s = \frac{dV}{dt}
\]
2. **Normal Acceleration (\( a_n \))**:
The normal acceleration for a curved path is given by:
\[
a_n = \frac{V^2}{r}
\]
where \( V \) is the velocity at \( t = 10 \) seconds and \( r \) is the radius of curvature of the pipe.
For this problem, apply the given velocity function \( V = 3t \) m/sec.
---
**Calculation Steps:**
1. **Find the Streamline Acceleration (\( a_s \)):**
\[
V = 3t \implies \frac{dV}{dt} = 3 \quad \text{m/s}^2
\]
At \( t = 10 \) sec, the streamline acceleration \( a_s \) is \( 3 \, \text{m/s}^2 \).
2. **Calculate Velocity at \( t = 10 \) seconds:**
\[
V = 3 \times 10 = 30 \, \text{m/s}
\]
3. **Find the Normal Acceleration (\( a_n \)):**
\[
a_n = \frac{V^2}{r} = \frac{30^2}{0.5} = \frac{900}{0.5} =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93c89597-7ce2-4451-bd61-6716aa1c2201%2F3d9fe282-2a4c-44d9-a55a-892b91128810%2Frkh7su_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
A fluid flows through a fixed (i.e., not rotating) curved pipe of uniform diameter 15 cm with a velocity of the fluid particles on the streamline along the center line of the pipe defined by \( V = 3t \) m/sec. Determine the normal acceleration and the streamline acceleration when \( t = 10 \) sec.
**Diagram Explanation:**
The diagram shows a curved pipe with a uniform diameter. The fluid flows from one end to the other along a streamline. The curvature of the pipe has a radius of 0.5 m. The direction of the fluid flow is indicated by arrows.
---
**Instructions on How to Solve:**
To determine the normal and streamline acceleration at \( t = 10 \) seconds, use the following formulas:
1. **Streamline Acceleration (\( a_s \))**:
The streamline acceleration is the derivative of the velocity function with respect to time \( t \).
\[
a_s = \frac{dV}{dt}
\]
2. **Normal Acceleration (\( a_n \))**:
The normal acceleration for a curved path is given by:
\[
a_n = \frac{V^2}{r}
\]
where \( V \) is the velocity at \( t = 10 \) seconds and \( r \) is the radius of curvature of the pipe.
For this problem, apply the given velocity function \( V = 3t \) m/sec.
---
**Calculation Steps:**
1. **Find the Streamline Acceleration (\( a_s \)):**
\[
V = 3t \implies \frac{dV}{dt} = 3 \quad \text{m/s}^2
\]
At \( t = 10 \) sec, the streamline acceleration \( a_s \) is \( 3 \, \text{m/s}^2 \).
2. **Calculate Velocity at \( t = 10 \) seconds:**
\[
V = 3 \times 10 = 30 \, \text{m/s}
\]
3. **Find the Normal Acceleration (\( a_n \)):**
\[
a_n = \frac{V^2}{r} = \frac{30^2}{0.5} = \frac{900}{0.5} =
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