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Three particles of mass m, 4m and 4m are constrained to move along a frictionless horizontal groove. The first two particles are connected by a spring of stiffness k and natural length lo. The last two particles are connected by a spring of stiffness 2k and natural length lo. Initially the particles are in equilibrium at points A, B and C and we measure the displacements x, y and z of the three particles from these equilibrium points, as shown in the following diagram. m Ax k, lo mrrrrrrrrr The system is displaced from equilibrium and we are concerned with modelling the subsequent motion. (a) Draw a force diagram for each particle, showing all of the forces acting on the particles. 4m 2k, lo 4m mm m m m m m m m m (b) Determine the changes in spring forces from their equilibrium values in terms of the above variables and parameters. (c) Derive the equation of motion of this mechanical system in the form (d) Show that T BY (1) where A is a matrix that you should specify explicitly. () 1 is an eigenvector of A by calculating the = A I y Z (e) Show that corresponding eigenvalue. What is the physical significance of this eigenvector in terms of the motion of the particles? (-).. -2 is an eigenvector of A by calculating the corresponding eigenvalue. (f) Use the trace formula to calculate a third eigenvalue of A and then calculate the corresponding eigenvector. E (g) Hence write down the solution to the equation of motion for this system.

Three particles of mass m, 4m and 4m are constrained to move along a frictionless horizontal groove. The first two particles are connected by a spring of stiffness k and natural length lo. The last two particles are connected by a spring of stiffness 2k and natural length lo. Initially the particles are in equilibrium at points A, B and C and we measure the displacements x, y and z of the three particles from these equilibrium points, as shown in the following diagram. m Ax k, lo mrrrrrrrrr The system is displaced from equilibrium and we are concerned with modelling the subsequent motion. (a) Draw a force diagram for each particle, showing all of the forces acting on the particles. 4m 2k, lo 4m mm m m m m m m m m (b) Determine the changes in spring forces from their equilibrium values in terms of the above variables and parameters. (c) Derive the equation of motion of this mechanical system in the form (d) Show that T BY (1) where A is a matrix that you should specify explicitly. () 1 is an eigenvector of A by calculating the = A I y Z (e) Show that corresponding eigenvalue. What is the physical significance of this eigenvector in terms of the motion of the particles? (-).. -2 is an eigenvector of A by calculating the corresponding eigenvalue. (f) Use the trace formula to calculate a third eigenvalue of A and then calculate the corresponding eigenvector. E (g) Hence write down the solution to the equation of motion for this system.

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
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Question 27 Three particles of mass m, 4m and 4m are constrained to move along a frictionless horizontal groove. The first two particles are connected by a spring of stiffness k and natural length lo. The last two particles are connected by a spring of stiffness 2k and natural length lo. Initially the particles are in equilibrium at points A, B and C and we measure the displacements x, y and z of the three particles from these equilibrium points, as shown in the following diagram. m k, lo mmmmmmmmmm 4m 2k, lo mo o o o o o o o o o A * The system is displaced from equilibrium and we are concerned with modelling the subsequent motion. 4m BY (a) Draw a force diagram for each particle, showing all of the forces acting on the particles. (d) Show that C² (b) Determine the changes in spring forces from their equilibrium values in terms of the above variables and parameters. (c) Derive the equation of motion of this mechanical system in the form ()--() = A y where A is a matrix that you should specify explicitly. () 1 is an eigenvector of A by calculating the (e) Show that corresponding eigenvalue. What is the physical significance of this eigenvector in terms of the motion of the particles? (-). corresponding eigenvalue. (f) Use the trace formula to calculate a third eigenvalue of A and then calculate the corresponding eigenvector. is an eigenvector of A by calculating the (g) Hence write down the solution to the equation of motion for this system.
Transcribed Image Text:Question 27 Three particles of mass m, 4m and 4m are constrained to move along a frictionless horizontal groove. The first two particles are connected by a spring of stiffness k and natural length lo. The last two particles are connected by a spring of stiffness 2k and natural length lo. Initially the particles are in equilibrium at points A, B and C and we measure the displacements x, y and z of the three particles from these equilibrium points, as shown in the following diagram. m k, lo mmmmmmmmmm 4m 2k, lo mo o o o o o o o o o A * The system is displaced from equilibrium and we are concerned with modelling the subsequent motion. 4m BY (a) Draw a force diagram for each particle, showing all of the forces acting on the particles. (d) Show that C² (b) Determine the changes in spring forces from their equilibrium values in terms of the above variables and parameters. (c) Derive the equation of motion of this mechanical system in the form ()--() = A y where A is a matrix that you should specify explicitly. () 1 is an eigenvector of A by calculating the (e) Show that corresponding eigenvalue. What is the physical significance of this eigenvector in terms of the motion of the particles? (-). corresponding eigenvalue. (f) Use the trace formula to calculate a third eigenvalue of A and then calculate the corresponding eigenvector. is an eigenvector of A by calculating the (g) Hence write down the solution to the equation of motion for this system.
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