### Understanding the Conservation of Energy in a System with Conservative Forces#### Problem Statement:When only conservative forces act on a system, the conservation of energy theorem can be utilized to solve problems effectively. For particle kinetics, the principle is expressed as:\[ T_1 + V_1 = T_2 + V_2 \]where \( T \) represents kinetic energy and \( V \) signifies potential energy, comprising gravitational potential energy (\( V_g \)) and elastic potential energy (\( V_e \)).#### Given Data:- The potential energy contributions are \( V_g = Wy_c \) (gravitational potential energy) and \( V_e = \frac{1}{2} k s^2 \) (elastic potential energy).- A rod with a mass of 30 kg is released from rest at an angle \(\theta = 45^\circ\).- The spring is unstretched when \(\theta = 45^\circ\).- The goal is to determine the angular velocity of the rod when \(\theta = 0^\circ\) using the conservation of energy theorem.#### System Description:- The rod \( AB \) is 1.5 meters in length.- The spring constant \( k \) is \( 300 \, \text{N/m} \).- The initial conditions (position and rest state) are specified for \(\theta = 45^\circ\).### Diagram Explanation:The accompanying diagram shows:- A rod \( AB \) attached to a fixed point at \( B \).- An unstretched spring connected to the rod at point \( A \) when \(\theta = 45^\circ\).- The rod is positioned at an angle \(\theta\) from the horizontal, initially set to \( 45^\circ\).- The length of the rod is marked as 1.5 meters, illustrating its pivot and the extension mechanism of the spring as the angle changes.### Objective:By leveraging the conservation of energy theorem, one can find the angular velocity of the rod when \(\theta\) is \(0^\circ\). This process involves:1. Calculating the initial potential energy due to both gravity and spring elasticity when \(\theta = 45^\circ\).2. Determining the final potential energy at \(\theta = 0^\circ\).3. Using the difference in potential energies to uphold the conservation of energy principle and

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Answered: determine the angular velocity of the…

Engineering

Mechanical Engineering

determine the angular velocity of the rod when 0 = 0°.

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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Concept explainers

Conservation of Energy

In physics, the rule of the conservation of energy states that energy doesn’t disappear from the atmosphere; instead, it changes its form from one type to another. Energy is one of the most important physical quantities in nature, and it cannot be created or destroyed — only transformed.

Question

### Understanding the Conservation of Energy in a System with Conservative Forces

#### Problem Statement:
When only conservative forces act on a system, the conservation of energy theorem can be utilized to solve problems effectively. For particle kinetics, the principle is expressed as:

\[ T_1 + V_1 = T_2 + V_2 \]

where \( T \) represents kinetic energy and \( V \) signifies potential energy, comprising gravitational potential energy (\( V_g \)) and elastic potential energy (\( V_e \)).

#### Given Data:
- The potential energy contributions are \( V_g = Wy_c \) (gravitational potential energy) and \( V_e = \frac{1}{2} k s^2 \) (elastic potential energy).
- A rod with a mass of 30 kg is released from rest at an angle \(\theta = 45^\circ\).
- The spring is unstretched when \(\theta = 45^\circ\).
- The goal is to determine the angular velocity of the rod when \(\theta = 0^\circ\) using the conservation of energy theorem.

#### System Description:
- The rod \( AB \) is 1.5 meters in length.
- The spring constant \( k \) is \( 300 \, \text{N/m} \).
- The initial conditions (position and rest state) are specified for \(\theta = 45^\circ\).

### Diagram Explanation:
The accompanying diagram shows:
- A rod \( AB \) attached to a fixed point at \( B \).
- An unstretched spring connected to the rod at point \( A \) when \(\theta = 45^\circ\).
- The rod is positioned at an angle \(\theta\) from the horizontal, initially set to \( 45^\circ\).
- The length of the rod is marked as 1.5 meters, illustrating its pivot and the extension mechanism of the spring as the angle changes.

### Objective:
By leveraging the conservation of energy theorem, one can find the angular velocity of the rod when \(\theta\) is \(0^\circ\).

This process involves:
1. Calculating the initial potential energy due to both gravity and spring elasticity when \(\theta = 45^\circ\).
2. Determining the final potential energy at \(\theta = 0^\circ\).
3. Using the difference in potential energies to uphold the conservation of energy principle and

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