Homework Help is Here – Start Your Trial Now!
Engineering
Mechanical Engineering
Coupled Harmonic Oscillators X, =0 x= x' = 2t Derivative X= x" = 2nd Derivative ki k2 k3 we M A) Write down the 2nd law for each of the masses. Use coordinates x, and x2 for m and M, and % X, and a,92.

Coupled Harmonic Oscillators X, =0 x= x' = 2t Derivative X= x" = 2nd Derivative ki k2 k3 we M A) Write down the 2nd law for each of the masses. Use coordinates x, and x2 for m and M, and % X, and a,92.

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
See similar textbooks
icon
Related questions
icon
Concept explainers
Question

please show all work for parts A-C. :)

Coupled Harmonic Oscillators X, 30 k3 M *= x' = 2t Derivative X= x" = 2nd Derivative ki wee A) Write down the 2nd law for each of the masses. Use coordinates x. and xz for m and M, and *, X2 and a,,a2. Hìnt : the force from spring I on m only depends on X,, but the force from spring 2 on m depends on (x2-x1). B) To simplify, let k, = k3 =k and m= M. Rownite cquations from parrt A) right abore each other. C) Define two new variabks , X( Greek "chi") =x. +xa and AX =x-X2. Then add and subtract your two differential evahions to produce two new, but very simple (harmonic) ones. D) Write down the solution to both egvations using wo=/m, wa=ktak. and coefficients A, B,C, and D. E) Now assume k=l0k2 (this means that the two masses are "weakly coupled"). Also assume x,(0) = -10cm, i,l0) =0, X. (0) =0, x1 (0) = 0. Solve for A, B,C,D, and solve for x,lt). F) Look up a trig. identity to show that your solution from part E) can be two differential your written in the form : x,(t) = = -10cm cos ( Wotwat Cas W2- 2 G) For k= loN/m, m= 1.00 kg graph the behavior of x,(t) from t=0 to t= 1T W2-Wo H) Based on your answer to part W guess what x(t) looks like. Try to guess X2(t)'s functional form.
Transcribed Image Text:Coupled Harmonic Oscillators X, 30 k3 M *= x' = 2t Derivative X= x" = 2nd Derivative ki wee A) Write down the 2nd law for each of the masses. Use coordinates x. and xz for m and M, and *, X2 and a,,a2. Hìnt : the force from spring I on m only depends on X,, but the force from spring 2 on m depends on (x2-x1). B) To simplify, let k, = k3 =k and m= M. Rownite cquations from parrt A) right abore each other. C) Define two new variabks , X( Greek "chi") =x. +xa and AX =x-X2. Then add and subtract your two differential evahions to produce two new, but very simple (harmonic) ones. D) Write down the solution to both egvations using wo=/m, wa=ktak. and coefficients A, B,C, and D. E) Now assume k=l0k2 (this means that the two masses are "weakly coupled"). Also assume x,(0) = -10cm, i,l0) =0, X. (0) =0, x1 (0) = 0. Solve for A, B,C,D, and solve for x,lt). F) Look up a trig. identity to show that your solution from part E) can be two differential your written in the form : x,(t) = = -10cm cos ( Wotwat Cas W2- 2 G) For k= loN/m, m= 1.00 kg graph the behavior of x,(t) from t=0 to t= 1T W2-Wo H) Based on your answer to part W guess what x(t) looks like. Try to guess X2(t)'s functional form.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps with 1 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Forming and Shaping
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Elements Of Electromagnetics
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Control Systems Engineering
Control Systems Engineering
Mechanical Engineering
ISBN:
9781118170519
Author:
Norman S. Nise
Publisher:
WILEY
Mechanics of Materials (MindTap Course List)
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:
9781337093347
Author:
Barry J. Goodno, James M. Gere
Publisher:
Cengage Learning
Engineering Mechanics: Statics
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:
9781118807330
Author:
James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:
WILEY
SEE MORE TEXTBOOKS