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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
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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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)
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)
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