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Starting from rest when S=0 the shaft of the motor M turns with an angular acceleration a=3 rad/s² Determine the speed of block W when it has risen S=6 m. 300 mm

Starting from rest when S=0 the shaft of the motor M turns with an angular acceleration a=3 rad/s² Determine the speed of block W when it has risen S=6 m. 300 mm

Elements Of Electromagnetics
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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**Determining the Speed of Block W Under Angular Acceleration** **Problem Statement** Starting from rest when \( S = 0 \), the shaft of motor \( M \) turns with an angular acceleration of \( \alpha = 3 \, \text{rad/s}^2 \). Determine the speed of block \( W \) when it has risen \( S = 6 \, \text{m} \). **Diagram Description** The provided diagram features a detailed gearing system driven by motor \( M \). The motor \( M \) rotates a shaft connected to gear \( A \) with a radius of 40 mm. Gear \( A \) is in mesh with a larger gear \( B \) of radius 225 mm. Gear \( B \) is fixed to gear \( C \), which has the same radius as \( B \). Gear \( C \) rotates another gear \( D \), which has a radius of 300 mm. Gear \( D \) is fixed to a drum \( E \), which winds a rope connected to block \( W \). The radius of drum \( E \) equals the radius of gear \( D \), i.e., 50 mm. **Analysis Steps** 1. **Gear Relationships:** - The relationship between the gears can be established using the ratios of their radii. - Acting through gears \( A \) to \( D \), the speed transformation must be calculated step-by-step. 2. **Angular Acceleration Transfer:** - Gears connected directly will share angular velocities inversely proportional to their radii. 3. **Linear Speed of Block W:** - The linear speed of block \( W \) is obtained by considering the angular velocity of drum \( E \). **Summary of Gear Radii:** - \( r_A = 40 \, \text{mm} \) - \( r_B = 225 \, \text{mm} \) - \( r_C = 225 \, \text{mm} \) - \( r_D = 300 \, \text{mm} \) - \( r_E = 50 \, \text{mm} \) **Solution Steps:** 1. Determine the angular velocity of each gear based on given accelerations. 2. Convert the resulting angular velocity to the linear velocity of the rope and effectively of block \( W \) by considering the motion derivations through the gears and finally
Transcribed Image Text:**Determining the Speed of Block W Under Angular Acceleration** **Problem Statement** Starting from rest when \( S = 0 \), the shaft of motor \( M \) turns with an angular acceleration of \( \alpha = 3 \, \text{rad/s}^2 \). Determine the speed of block \( W \) when it has risen \( S = 6 \, \text{m} \). **Diagram Description** The provided diagram features a detailed gearing system driven by motor \( M \). The motor \( M \) rotates a shaft connected to gear \( A \) with a radius of 40 mm. Gear \( A \) is in mesh with a larger gear \( B \) of radius 225 mm. Gear \( B \) is fixed to gear \( C \), which has the same radius as \( B \). Gear \( C \) rotates another gear \( D \), which has a radius of 300 mm. Gear \( D \) is fixed to a drum \( E \), which winds a rope connected to block \( W \). The radius of drum \( E \) equals the radius of gear \( D \), i.e., 50 mm. **Analysis Steps** 1. **Gear Relationships:** - The relationship between the gears can be established using the ratios of their radii. - Acting through gears \( A \) to \( D \), the speed transformation must be calculated step-by-step. 2. **Angular Acceleration Transfer:** - Gears connected directly will share angular velocities inversely proportional to their radii. 3. **Linear Speed of Block W:** - The linear speed of block \( W \) is obtained by considering the angular velocity of drum \( E \). **Summary of Gear Radii:** - \( r_A = 40 \, \text{mm} \) - \( r_B = 225 \, \text{mm} \) - \( r_C = 225 \, \text{mm} \) - \( r_D = 300 \, \text{mm} \) - \( r_E = 50 \, \text{mm} \) **Solution Steps:** 1. Determine the angular velocity of each gear based on given accelerations. 2. Convert the resulting angular velocity to the linear velocity of the rope and effectively of block \( W \) by considering the motion derivations through the gears and finally
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