The 5-lb collar slides on the smooth rod, so that when it is at 4 it has a speed of 10 ft/s. If the spring to which it is attached has an unstretched length of 3 ft. and a stiffness of k = 10 lb/ft, determine the normal force on the collar and the acceleration of the collar at this instant. Solution: 2 ft 10 ft/s
The 5-lb collar slides on the smooth rod, so that when it is at 4 it has a speed of 10 ft/s. If the spring to which it is attached has an unstretched length of 3 ft. and a stiffness of k = 10 lb/ft, determine the normal force on the collar and the acceleration of the collar at this instant. Solution: 2 ft 10 ft/s
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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![**Problem Statement:**
The 5-lb collar slides on the smooth rod, so that when it is at A it has a speed of 10 ft/s. If the spring to which it is attached has an unstretched length of 3 ft and a stiffness of \( k = 10 \text{ lb/ft} \), determine the normal force on the collar and the acceleration of the collar at this instant.
**Diagram Description:**
The diagram shows a vertical rod OA with a smooth surface. A collar is sliding on the rod and is connected to a spring. The spring is attached to a point on the horizontal x-axis which is 2 ft away from the origin O.
The point A, where the collar is located, is on a curved path defined by the equation:
\[ y = 8 - \frac{1}{2}x^2 \]
At point A, the collar has a speed of 10 ft/s, and the spring is attached at this point as well, forming an angle with the rod.
The curved path and the spring setup create a dynamic system where forces and motion need to be analyzed.
**Given:**
- Weight of the collar: 5 lb
- Speed of the collar at A: 10 ft/s
- Unstretched length of the spring: 3 ft
- Spring stiffness: \( k = 10 \text{ lb/ft} \)
**Required:**
- Determine the normal force on the collar.
- Determine the acceleration of the collar at this instant.
**Solution Outline:**
1. Analyze the geometry of the setup to find the position of the collar and the spring extension.
2. Use dynamic equations of motion to determine the forces acting on the collar.
3. Calculate the acceleration and the normal force based on the spring force and the collar's motion.
This problem involves applying principles of dynamics, including kinematics and Newton's laws, to solve for the normal force and acceleration of the sliding collar.
### Detailed Steps:
1. **Determine Coordinates of A:**
Using the given curve equation, find the coordinates of point A.
2. **Compute Spring Extension:**
Calculate how much the spring is stretched or compressed from its natural length.
3. **Apply Newton's Second Law:**
Decompose the forces acting on the collar into tangential and normal components.
4. **Calculate Normal Force:**
Balance the forces perpendicular to the motion path to find](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5386f0c-0c7d-4f0c-9c77-b0195c82b8c2%2F80aa9a5e-e55a-46e8-9c3e-a197c379f110%2Fnjn4zok_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
The 5-lb collar slides on the smooth rod, so that when it is at A it has a speed of 10 ft/s. If the spring to which it is attached has an unstretched length of 3 ft and a stiffness of \( k = 10 \text{ lb/ft} \), determine the normal force on the collar and the acceleration of the collar at this instant.
**Diagram Description:**
The diagram shows a vertical rod OA with a smooth surface. A collar is sliding on the rod and is connected to a spring. The spring is attached to a point on the horizontal x-axis which is 2 ft away from the origin O.
The point A, where the collar is located, is on a curved path defined by the equation:
\[ y = 8 - \frac{1}{2}x^2 \]
At point A, the collar has a speed of 10 ft/s, and the spring is attached at this point as well, forming an angle with the rod.
The curved path and the spring setup create a dynamic system where forces and motion need to be analyzed.
**Given:**
- Weight of the collar: 5 lb
- Speed of the collar at A: 10 ft/s
- Unstretched length of the spring: 3 ft
- Spring stiffness: \( k = 10 \text{ lb/ft} \)
**Required:**
- Determine the normal force on the collar.
- Determine the acceleration of the collar at this instant.
**Solution Outline:**
1. Analyze the geometry of the setup to find the position of the collar and the spring extension.
2. Use dynamic equations of motion to determine the forces acting on the collar.
3. Calculate the acceleration and the normal force based on the spring force and the collar's motion.
This problem involves applying principles of dynamics, including kinematics and Newton's laws, to solve for the normal force and acceleration of the sliding collar.
### Detailed Steps:
1. **Determine Coordinates of A:**
Using the given curve equation, find the coordinates of point A.
2. **Compute Spring Extension:**
Calculate how much the spring is stretched or compressed from its natural length.
3. **Apply Newton's Second Law:**
Decompose the forces acting on the collar into tangential and normal components.
4. **Calculate Normal Force:**
Balance the forces perpendicular to the motion path to find
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