Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Transcribed Image Text:### Problem 18-57: Angular Velocity of a Rod Attached to a Spring
**Description:**
A spring with a stiffness of \( k = 300 \, \text{N/m} \) is connected to the end of a 15-kg rod. The rod is positioned vertically and the spring is unstretched when the angle \( \theta = 0^\circ \). The rod is released from rest at this position. The objective is to find the angular velocity of the rod when the angle \( \theta = 30^\circ \). The motion of the rod is confined to a vertical plane.
**Diagram Explanation:**
The diagram shows a rod of length \( 0.6 \, \text{m} \), pinned at point \( A \) and free to rotate. A spring of stiffness \( k = 300 \, \text{N/m} \) is attached at one end to point \( A \) and at the other end to point \( B \), which is also attached to the rod. The spring is vertical when the rod is horizontal.
- **Points A and B:**
- \( A \) is the pivot point where the rod is fixed.
- \( B \) is the point on the rod where the spring is attached.
- **Spring:**
- The spring is unstretched when \( \theta = 0^\circ \) (horizontal position).
- The spring's stiffness is \( k = 300 \, \text{N/m} \).
- **Rod Length:**
- The length of the rod from point \( A \) to point \( B \) is \( 0.6 \, \text{m} \).
- **Angle \( \theta \):**
- The rod makes an angle \( \theta \) with the horizontal.
**Problem Statement:**
Given that the rod is released from rest at \( \theta = 0^\circ \), determine the angular velocity when \( \theta = 30^\circ \).
**Concepts Involved:**
- **Potential Energy in the Spring:** When the rod is released and starts to rotate, the spring stretches, storing potential energy.
- **Kinetic Energy of the Rod:** As the rod moves, it gains kinetic energy which includes rotational kinetic energy due to its mass and angular velocity.
- **Conservation of Energy:** The total mechanical energy (sum of potential energy and
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