2/186 The hoisting system shown is used to easily raise kayaks for overhead storage. Determine expressions for the upward velocity and acceleration of the kayak at any height y if the winch M reels in cable at a constant rate i. As- sume that the kayak remains level.
2/186 The hoisting system shown is used to easily raise kayaks for overhead storage. Determine expressions for the upward velocity and acceleration of the kayak at any height y if the winch M reels in cable at a constant rate i. As- sume that the kayak remains level.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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I have included the problem and the solution for velocity and acceleration. Please show all steps to get to that solution. Thanks
![### Detailed Explanation of the Formulas
In this section, we present two mathematical formulas which are fundamental for understanding a particular aspect of kinematic or dynamic systems.
#### Formula 1: Velocity Equation
The first equation determines the velocity \( v \):
\[
v = \frac{i \sqrt{4y^2 + b^2}}{16y}
\]
- **\( i \)** is a parameter whose specific meaning depends on the context of the problem.
- **\( y \)** and **\( b \)** are variables that could represent dimensions such as distance or other quantities in the system being analyzed.
- The numerator \( i \sqrt{4y^2 + b^2} \) involves the square root of \( 4y^2 + b^2 \), which is then multiplied by \( i \).
- The denominator is \( 16y \), indicating a division of the entire expression by this term.
#### Formula 2: Acceleration Equation
The second equation calculates the acceleration \( a \):
\[
a = \frac{b^2 i^2}{256y^3}
\]
- **\( b^2 \)** is the square of \( b \), another parameter likely related to the physical or geometrical properties of the system.
- **\( i^2 \)** is the square of \( i \), showing it has a multiplicative role in this context.
- The denominator \( 256y^3 \) involves \( y \) raised to the power of 3, scaled by the factor 256, indicating a more rapid diminishing effect as \( y \) increases.
### Conceptual Understanding
To effectively use these formulas:
1. **Identify the Physical Context**: Understand the significance of \( i \), \( y \), and \( b \) in the scenario. They could represent specific measures like lengths, time, or other physical quantities.
2. **Evaluate the Parameters**: Substitute the known values or experimental data for \( i \), \( y \), and \( b \) to compute the velocity \( v \) and acceleration \( a \).
3. **Analyze the Relationships**: Notice how changes in \( y \) and \( b \) affect \( v \) and \( a \). For instance, increasing \( y \) in the velocity formula reduces \( v \) due to the division by \( 16y \).
Using these relationships can be crucial](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda0a2617-650f-47df-ae27-c9852e282155%2F1f18346d-cc9f-4478-8e8e-a156e9c210b2%2Fyz795re_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Detailed Explanation of the Formulas
In this section, we present two mathematical formulas which are fundamental for understanding a particular aspect of kinematic or dynamic systems.
#### Formula 1: Velocity Equation
The first equation determines the velocity \( v \):
\[
v = \frac{i \sqrt{4y^2 + b^2}}{16y}
\]
- **\( i \)** is a parameter whose specific meaning depends on the context of the problem.
- **\( y \)** and **\( b \)** are variables that could represent dimensions such as distance or other quantities in the system being analyzed.
- The numerator \( i \sqrt{4y^2 + b^2} \) involves the square root of \( 4y^2 + b^2 \), which is then multiplied by \( i \).
- The denominator is \( 16y \), indicating a division of the entire expression by this term.
#### Formula 2: Acceleration Equation
The second equation calculates the acceleration \( a \):
\[
a = \frac{b^2 i^2}{256y^3}
\]
- **\( b^2 \)** is the square of \( b \), another parameter likely related to the physical or geometrical properties of the system.
- **\( i^2 \)** is the square of \( i \), showing it has a multiplicative role in this context.
- The denominator \( 256y^3 \) involves \( y \) raised to the power of 3, scaled by the factor 256, indicating a more rapid diminishing effect as \( y \) increases.
### Conceptual Understanding
To effectively use these formulas:
1. **Identify the Physical Context**: Understand the significance of \( i \), \( y \), and \( b \) in the scenario. They could represent specific measures like lengths, time, or other physical quantities.
2. **Evaluate the Parameters**: Substitute the known values or experimental data for \( i \), \( y \), and \( b \) to compute the velocity \( v \) and acceleration \( a \).
3. **Analyze the Relationships**: Notice how changes in \( y \) and \( b \) affect \( v \) and \( a \). For instance, increasing \( y \) in the velocity formula reduces \( v \) due to the division by \( 16y \).
Using these relationships can be crucial
![### Problem 2/186: Hoisting System for Kayaks
#### Problem Statement:
The hoisting system shown is used to easily raise kayaks for overhead storage. Determine expressions for the upward velocity and acceleration of the kayak at any height \( y \) if the winch \( M \) reels in cable at a constant rate \( \dot{l} \). Assume that the kayak remains level.
#### Diagram Explanation:
The diagram illustrates a system for hoisting a kayak using a winch mechanism. The kayak is suspended by cables that are reeled in by winch \( M \). The setup includes three pulleys from which the kayak is suspended:
1. The winch \( M \) pulls the cable upward.
2. The kayak is connected at two points, labeled \( A \) and \( B \).
3. The distance between points \( A \) and \( B \) is denoted as \( 2b \).
The coordinates and distances of interest are as follows:
- The vertical distance from the pulleys to the kayak is represented as \( y \).
- The horizontal distances between the pulleys and the attachment points on the kayak are equal and denoted as \( b \).
### Key Concepts to Consider:
1. **Cable Length Changes:** Since the winch reels in the cable at a constant rate, the changing lengths of the cables will alter the height \( y \) of the kayak.
2. **Velocity and Acceleration:** As the cable is hoisted, the system’s mechanics (length of cable pulled in per time unit and resulting speed of kayak movement) need to be explored.
### Steps to Determine Upward Velocity and Acceleration:
1. **Total Length of Cable:**
Let \( L \) denote the total length of the cable. Since the cable forms a triangular shape with the vertical height \( y \) and horizontal distance \( b \), the length of each side of the triangle (hypotenuse) is given by \( \sqrt{b^2 + y^2} \).
2. **Total Cable on Both Sides:**
The total length of the cable from the kayak to the top is frequently used and is given by:
\[
L = 2\sqrt{b^2 + y^2}
\]
3. **Rate of Length Change:**
Given the winch reels in the cable at a constant rate \( \dot{l} \), the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda0a2617-650f-47df-ae27-c9852e282155%2F1f18346d-cc9f-4478-8e8e-a156e9c210b2%2Fbiifpat_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 2/186: Hoisting System for Kayaks
#### Problem Statement:
The hoisting system shown is used to easily raise kayaks for overhead storage. Determine expressions for the upward velocity and acceleration of the kayak at any height \( y \) if the winch \( M \) reels in cable at a constant rate \( \dot{l} \). Assume that the kayak remains level.
#### Diagram Explanation:
The diagram illustrates a system for hoisting a kayak using a winch mechanism. The kayak is suspended by cables that are reeled in by winch \( M \). The setup includes three pulleys from which the kayak is suspended:
1. The winch \( M \) pulls the cable upward.
2. The kayak is connected at two points, labeled \( A \) and \( B \).
3. The distance between points \( A \) and \( B \) is denoted as \( 2b \).
The coordinates and distances of interest are as follows:
- The vertical distance from the pulleys to the kayak is represented as \( y \).
- The horizontal distances between the pulleys and the attachment points on the kayak are equal and denoted as \( b \).
### Key Concepts to Consider:
1. **Cable Length Changes:** Since the winch reels in the cable at a constant rate, the changing lengths of the cables will alter the height \( y \) of the kayak.
2. **Velocity and Acceleration:** As the cable is hoisted, the system’s mechanics (length of cable pulled in per time unit and resulting speed of kayak movement) need to be explored.
### Steps to Determine Upward Velocity and Acceleration:
1. **Total Length of Cable:**
Let \( L \) denote the total length of the cable. Since the cable forms a triangular shape with the vertical height \( y \) and horizontal distance \( b \), the length of each side of the triangle (hypotenuse) is given by \( \sqrt{b^2 + y^2} \).
2. **Total Cable on Both Sides:**
The total length of the cable from the kayak to the top is frequently used and is given by:
\[
L = 2\sqrt{b^2 + y^2}
\]
3. **Rate of Length Change:**
Given the winch reels in the cable at a constant rate \( \dot{l} \), the
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