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1. In lecture I focused on a horizontal mass-spring system so we could ignore gravity, but in practice it's much easier to build a vertical mass-spring system. In the figure below, we can see a suspended spring with spring constant k both before and after a mass m is hooked to it. y yo is the equilibrium height of the spring (without the mass) and in the figure y = 0 is set at the ground. a) Using the coordinate system in the figure, show that Newton's second law for the hanging mass-spring system leads to the following differential equation: Yo E = d²y k dt² (y-yo) - 9 m b) Now define a new coordinate, y', related to y by a constant shift: y = y' + C, where C = constant. Show that if you choose the right value of C, Newton's second law is identical in form to a mass-spring system without gravity. What is the period of the system? c) Effectively, gravity just shifts the equilibrium position of the system, but the system still undergoes simple harmonic motion. What is the net force on the mass when y' = 0? How high above the ground is the new equilibrium position in terms of yo, m, g, and k? d) Using the coordinate system in the figure, the total mechanical energy E of the mass-spring system is == 1 2 mv² +/-k(y − yo)² +mgy 2 Using the same change of coordinates as part b), show that up to an irrelevant constant the energy looks like a mass-spring system without gravity. e) What are y' and y as functions of time? Don't worry about determining any arbitrary constants, just very broadly write down their general forms.

1. In lecture I focused on a horizontal mass-spring system so we could ignore gravity, but in practice it's much easier to build a vertical mass-spring system. In the figure below, we can see a suspended spring with spring constant k both before and after a mass m is hooked to it. y yo is the equilibrium height of the spring (without the mass) and in the figure y = 0 is set at the ground. a) Using the coordinate system in the figure, show that Newton's second law for the hanging mass-spring system leads to the following differential equation: Yo E = d²y k dt² (y-yo) - 9 m b) Now define a new coordinate, y', related to y by a constant shift: y = y' + C, where C = constant. Show that if you choose the right value of C, Newton's second law is identical in form to a mass-spring system without gravity. What is the period of the system? c) Effectively, gravity just shifts the equilibrium position of the system, but the system still undergoes simple harmonic motion. What is the net force on the mass when y' = 0? How high above the ground is the new equilibrium position in terms of yo, m, g, and k? d) Using the coordinate system in the figure, the total mechanical energy E of the mass-spring system is == 1 2 mv² +/-k(y − yo)² +mgy 2 Using the same change of coordinates as part b), show that up to an irrelevant constant the energy looks like a mass-spring system without gravity. e) What are y' and y as functions of time? Don't worry about determining any arbitrary constants, just very broadly write down their general forms.

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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1. In lecture I focused on a horizontal mass-spring system so we could ignore gravity, but in practice it's much easier to build a vertical mass-spring system. In the figure below, we can see a suspended spring with spring constant k both before and after a mass m is hooked to it. Y Yo yo is the equilibrium height of the spring (without the mass) and in the figure y = 0 is set at the ground. a) Using the coordinate system in the figure, show that Newton's second law for the hanging mass-spring system leads to the following differential equation: d²y dt² k (y - Yo) - g m b) Now define a new coordinate, y', related to y by a constant shift: y = y' + C, where C = constant. Show that if you choose the right value of C, Newton's second law is identical in form to a mass-spring system without gravity. What is the period of the system? c) Effectively, gravity just shifts the equilibrium position of the system, but the system still undergoes simple harmonic motion. What is the net force on the mass when y' = 0? How high above the ground is the new equilibrium position in terms of yo, m, g, and k? d) Using the coordinate system in the figure, the total mechanical energy E of the mass-spring system is 1 E = -mv² + 2k(y − yo)² + mgy Using the same change of coordinates as part b), show that up to an irrelevant constant the energy looks like a mass-spring system without gravity. e) What are y' and y as functions of time? Don't worry about determining any arbitrary constants, just very broadly write down their general forms.
Transcribed Image Text:1. In lecture I focused on a horizontal mass-spring system so we could ignore gravity, but in practice it's much easier to build a vertical mass-spring system. In the figure below, we can see a suspended spring with spring constant k both before and after a mass m is hooked to it. Y Yo yo is the equilibrium height of the spring (without the mass) and in the figure y = 0 is set at the ground. a) Using the coordinate system in the figure, show that Newton's second law for the hanging mass-spring system leads to the following differential equation: d²y dt² k (y - Yo) - g m b) Now define a new coordinate, y', related to y by a constant shift: y = y' + C, where C = constant. Show that if you choose the right value of C, Newton's second law is identical in form to a mass-spring system without gravity. What is the period of the system? c) Effectively, gravity just shifts the equilibrium position of the system, but the system still undergoes simple harmonic motion. What is the net force on the mass when y' = 0? How high above the ground is the new equilibrium position in terms of yo, m, g, and k? d) Using the coordinate system in the figure, the total mechanical energy E of the mass-spring system is 1 E = -mv² + 2k(y − yo)² + mgy Using the same change of coordinates as part b), show that up to an irrelevant constant the energy looks like a mass-spring system without gravity. e) What are y' and y as functions of time? Don't worry about determining any arbitrary constants, just very broadly write down their general forms.
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