### Problem #1#### System Description:Consider the system displayed in the diagram. The two pulleys, one small and one large, are connected together to move as a single unit. #### Diagram Explanation:- **Pulley System:** Two pulleys are shown—\( R_1, m_1 \) as the small pulley and \( R_2, m_2 \) as the large pulley.- **Mass and Spring-Damper System:** A mass \( m \) is attached to the system and its displacement is represented as \( x(t) \).- **Springs:** \( k_1 \) and \( k_2 \) are the spring constants.- **Dampers:** \( c_1 \) is a damping coefficient and \( c_2 \) is a design parameter that needs to be determined.#### Note:The inertia of the pulley system is given by: \[ J_{\text{disc}} = \frac{1}{2} mR^2 \]#### Parameter Values:- \( m = 5 \, \text{kg} \)- \( m_1 = 1 \, \text{kg} \)- \( m_2 = 2 \, \text{kg} \)- \( R_1 = 0.200 \, \text{m} \)- \( R_2 = 0.400 \, \text{m} \)- \( k_1 = 400 \, \text{N/m} \)- \( k_2 = 300 \, \text{N/m} \)- \( c_1 = 1.5 \, \text{N·s/m} \)- \( c_2 = \) design parameter#### Tasks:a. **Determine the characteristic equation** in terms of the position of the hanging mass, \( x(t) \).b. **Adjustable Parameter \( c_2 \):** In the design phase, determine the value of \( c_2 \) needed to achieve a damping coefficient of 0.2.c. **System Response in Post-Design Phase:** Calculate the system response, \( x(t) \), given initial conditions: - \( x(t = 0) = 0.01 \, \text{m} \) - \( \dot{x}(t = 0) = 0

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Consider the system shown below. The two pulleys, small and large, are bolted together and move as a single piece. The parameter values are as follows: m = 5 kg m₂ = 1 kg m₂ = 2 kg R₂, ma R₁.m R₁ = 0.200 m k₂ = 400 N/m k₂= 300 N/m E Note: Jaise == x(t = 0) = 0.01 m i(t = 0) = 0 MR² R₂ = 0.400 m C₁ = 1.5 N-s/m C₂= design parameter a. Determine the characteristic equation in terms of the position of the hanging mass, x(t). b. The parameter c, can be adjusted in the design phase. Determine the value of c, required to yield a damping coefficient of 0.2 c. In the post-design phase structure, determine the system response, x(t), in response to initial conditions of

Consider the system shown below. The two pulleys, small and large, are bolted together and move as a single piece. The parameter values are as follows: m = 5 kg m₂ = 1 kg m₂ = 2 kg R₂, ma R₁.m R₁ = 0.200 m k₂ = 400 N/m k₂= 300 N/m E Note: Jaise == x(t = 0) = 0.01 m i(t = 0) = 0 MR² R₂ = 0.400 m C₁ = 1.5 N-s/m C₂= design parameter a. Determine the characteristic equation in terms of the position of the hanging mass, x(t). b. The parameter c, can be adjusted in the design phase. Determine the value of c, required to yield a damping coefficient of 0.2 c. In the post-design phase structure, determine the system response, x(t), in response to initial conditions of

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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Question

### Problem #1

#### System Description:
Consider the system displayed in the diagram. The two pulleys, one small and one large, are connected together to move as a single unit.

#### Diagram Explanation:
- **Pulley System:** Two pulleys are shown—\( R_1, m_1 \) as the small pulley and \( R_2, m_2 \) as the large pulley.
- **Mass and Spring-Damper System:** A mass \( m \) is attached to the system and its displacement is represented as \( x(t) \).
- **Springs:** \( k_1 \) and \( k_2 \) are the spring constants.
- **Dampers:** \( c_1 \) is a damping coefficient and \( c_2 \) is a design parameter that needs to be determined.

#### Note:
The inertia of the pulley system is given by:
\[ J_{\text{disc}} = \frac{1}{2} mR^2 \]

#### Parameter Values:
- \( m = 5 \, \text{kg} \)
- \( m_1 = 1 \, \text{kg} \)
- \( m_2 = 2 \, \text{kg} \)
- \( R_1 = 0.200 \, \text{m} \)
- \( R_2 = 0.400 \, \text{m} \)
- \( k_1 = 400 \, \text{N/m} \)
- \( k_2 = 300 \, \text{N/m} \)
- \( c_1 = 1.5 \, \text{N·s/m} \)
- \( c_2 = \) design parameter

#### Tasks:

a. **Determine the characteristic equation** in terms of the position of the hanging mass, \( x(t) \).

b. **Adjustable Parameter \( c_2 \):** In the design phase, determine the value of \( c_2 \) needed to achieve a damping coefficient of 0.2.

c. **System Response in Post-Design Phase:** Calculate the system response, \( x(t) \), given initial conditions:
- \( x(t = 0) = 0.01 \, \text{m} \)
- \( \dot{x}(t = 0) = 0

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