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An elliptical hoop of negligible thickness with a vertical orientated semi-major axis a and a horizontal orientated semi-minor axis b, which has a mass per unit length of p, is mounted on a circular peg to which is attached a Carte- sian xy coordinate system in a vertical orienta- tion as shown in the accompanying sketch (the gravity g ellipse and peg are not rigidly joined to each other and are simply in kinematic contact with each other with negligible friction), such that the physical system lies within a flat xy plane. The ellipse is then allowed to freely oscillate in a to-and-fro rocking motion by a small angle 0, such that the gravitational acceleration g acts in a downwards direction that is parallel to the ellipse semi-major axis. 1.3 a Considering the above system determine the following information clearly showing all steps and appropriate reasoning: 1.4 Y Utilizing the previously derived equation of motion determine the corresponding natural angular frequency wn for the Simple Harmonic Motion (SHM) of the vibrating system; Explain what would happen to the engineering dynamics of the vibration system if the ellipse were in an initial configuration where the semi-major axis a was horizontal and the semi-minor axis b was vertical and why this would occur using appropriate mathematical and physical reasoning. I

An elliptical hoop of negligible thickness with a vertical orientated semi-major axis a and a horizontal orientated semi-minor axis b, which has a mass per unit length of p, is mounted on a circular peg to which is attached a Carte- sian xy coordinate system in a vertical orienta- tion as shown in the accompanying sketch (the gravity g ellipse and peg are not rigidly joined to each other and are simply in kinematic contact with each other with negligible friction), such that the physical system lies within a flat xy plane. The ellipse is then allowed to freely oscillate in a to-and-fro rocking motion by a small angle 0, such that the gravitational acceleration g acts in a downwards direction that is parallel to the ellipse semi-major axis. 1.3 a Considering the above system determine the following information clearly showing all steps and appropriate reasoning: 1.4 Y Utilizing the previously derived equation of motion determine the corresponding natural angular frequency wn for the Simple Harmonic Motion (SHM) of the vibrating system; Explain what would happen to the engineering dynamics of the vibration system if the ellipse were in an initial configuration where the semi-major axis a was horizontal and the semi-minor axis b was vertical and why this would occur using appropriate mathematical and physical reasoning. I

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
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An elliptical hoop of negligible thickness with a vertical orientated semi-major axis a and a horizontal orientated semi-minor axis b, which has a mass per unit length of p, is mounted on a circular peg to which is attached a Carte- sian xy coordinate system in a vertical orienta- tion as shown in the accompanying sketch (the gravity g ellipse and peg are not rigidly joined to each other and are simply in kinematic contact with each other with negligible friction), such that the physical system lies within a flat ry plane. The ellipse is then allowed to freely oscillate in a to-and-fro rocking motion by a small angle 0, such that the gravitational acceleration g acts in a downwards direction that is parallel to the ellipse semi-major axis. 1.3 a 1.4 b Considering the above system determine the following information clearly showing all steps and appropriate reasoning: Y Utilizing the previously derived equation of motion determine the corresponding natural angular frequency wn for the Simple Harmonic Motion (SHM) of the vibrating system; Explain what would happen to the engineering dynamics of the vibration system if the ellipse were in an initial configuration where the semi-major axis a was horizontal and the semi-minor axis b was vertical and why this would occur using appropriate mathematical and physical reasoning. x
Transcribed Image Text:An elliptical hoop of negligible thickness with a vertical orientated semi-major axis a and a horizontal orientated semi-minor axis b, which has a mass per unit length of p, is mounted on a circular peg to which is attached a Carte- sian xy coordinate system in a vertical orienta- tion as shown in the accompanying sketch (the gravity g ellipse and peg are not rigidly joined to each other and are simply in kinematic contact with each other with negligible friction), such that the physical system lies within a flat ry plane. The ellipse is then allowed to freely oscillate in a to-and-fro rocking motion by a small angle 0, such that the gravitational acceleration g acts in a downwards direction that is parallel to the ellipse semi-major axis. 1.3 a 1.4 b Considering the above system determine the following information clearly showing all steps and appropriate reasoning: Y Utilizing the previously derived equation of motion determine the corresponding natural angular frequency wn for the Simple Harmonic Motion (SHM) of the vibrating system; Explain what would happen to the engineering dynamics of the vibration system if the ellipse were in an initial configuration where the semi-major axis a was horizontal and the semi-minor axis b was vertical and why this would occur using appropriate mathematical and physical reasoning. x
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