A particle of mass 3m is suspended from the ceiling by a spring of force constant k. A second particle of mass 2m is suspended from the first particle by a second identical spring. The rest lengths of both springs are lo. The particles can only move vertically. Gravity acts as usual in the vertical direction. www 3m 2m ееее ееее k (a) How many degrees of freedom does the system have? (b) Choose appropriate generalised coordinates and write down the Lagrangian of the system in terms of them. (c) Write down the Euler-Lagrange equations (by this I mean the explicit equations, so do perform the derivatives of the Lagrangian). (d) Find the equilibrium position of the particles. Is it stable? Note: Remember our general three-step procedure for solving problems: 1. Find adequate generalised coordinates {q;}1 to describe the motion of the system. The number of these coordinates, M, is the number of degrees of freedom of the system. I will collect them in an M-dimensional vector, which I call q. 2. Re-express the kinetic energy T :=1 m/2 in terms of the generalised coordinates and momenta q and q (here n is the number of particles in the system). 3. Write the Lagrangian L(q, ġ) = T(q, ġ) - V (q). There will be one Lagrange equation for each gener- alised coordinate q; you have, corresponding to each degree of freedom. These equations are d ƏL dt oq = ƏL Əq (1)
A particle of mass 3m is suspended from the ceiling by a spring of force constant k. A second particle of mass 2m is suspended from the first particle by a second identical spring. The rest lengths of both springs are lo. The particles can only move vertically. Gravity acts as usual in the vertical direction. www 3m 2m ееее ееее k (a) How many degrees of freedom does the system have? (b) Choose appropriate generalised coordinates and write down the Lagrangian of the system in terms of them. (c) Write down the Euler-Lagrange equations (by this I mean the explicit equations, so do perform the derivatives of the Lagrangian). (d) Find the equilibrium position of the particles. Is it stable? Note: Remember our general three-step procedure for solving problems: 1. Find adequate generalised coordinates {q;}1 to describe the motion of the system. The number of these coordinates, M, is the number of degrees of freedom of the system. I will collect them in an M-dimensional vector, which I call q. 2. Re-express the kinetic energy T :=1 m/2 in terms of the generalised coordinates and momenta q and q (here n is the number of particles in the system). 3. Write the Lagrangian L(q, ġ) = T(q, ġ) - V (q). There will be one Lagrange equation for each gener- alised coordinate q; you have, corresponding to each degree of freedom. These equations are d ƏL dt oq = ƏL Əq (1)
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
Related questions
Question
Transcribed Image Text:Problem 2
A particle of mass 3m is suspended from the ceiling by a spring of force constant k. A second particle of
mass 2m is suspended from the first particle by a second identical spring. The rest lengths of both springs
are lo. The particles can only move vertically. Gravity acts as usual in the vertical direction.
3m
2m
ееееееее
k
k
(a) How many degrees of freedom does the system have?
(b) Choose appropriate generalised coordinates and write down the Lagrangian of the system in terms of
them.
(c) Write down the Euler-Lagrange equations (by this I mean the explicit equations, so do perform the
derivatives of the Lagrangian).
(d) Find the equilibrium position of the particles. Is it stable?
Note: Remember our general three-step procedure for solving problems:
1. Find adequate generalised coordinates {q} to describe the motion of the system. The number of
these coordinates, M, is the number of degrees of freedom of the system. I will collect them in an
M-dimensional vector, which I call q.
2. Re-express the kinetic energy T := Σ1 m₂/2 in terms of the generalised coordinates and momenta
q and q (here n is the number of particles in the system).
d ƏL
dt əq
3. Write the Lagrangian L(q, q) = T(q, q) - V(q). There will be one Lagrange equation for each gener-
alised coordinate q; you have, corresponding to each degree of freedom. These equations are
ƏL
iq
(1)
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Knowledge Booster
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
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Elements Of Electromagnetics
Mechanical Engineering
ISBN:
9780190698614
Author:
Sadiku, Matthew N. O.
Publisher:
Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:
9780134319650
Author:
Russell C. Hibbeler
Publisher:
PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:
9781259822674
Author:
Yunus A. Cengel Dr., Michael A. Boles
Publisher:
McGraw-Hill Education
Control Systems Engineering
Mechanical Engineering
ISBN:
9781118170519
Author:
Norman S. Nise
Publisher:
WILEY
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:
9781337093347
Author:
Barry J. Goodno, James M. Gere
Publisher:
Cengage Learning
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:
9781118807330
Author:
James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:
WILEY