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 mrrrrrrrrr 4m Ka 72 The system is displaced from equilibrium and we are concerned with modelling the subsequent motion. 2k, lo 4m mmmmmmmmmm BY (a) Draw a force diagram for each particle, showing all of the forces acting on the particles. (b) (c) Derive the equation of motion of this mechanical system in the form (d) Show that Determine the changes in spring forces from their equilibrium values in terms of the above variables and parameters. 0--0) = A where A is a matrix that you should specify explicitly. (9) > is an eigenvector of A by calculating the corresponding eigenvalue. What is the physical significance of this eigenvector in terms of the motion of the particles?

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 mrrrrrrrrr 4m Ka 72 The system is displaced from equilibrium and we are concerned with modelling the subsequent motion. 2k, lo 4m mmmmmmmmmm BY (a) Draw a force diagram for each particle, showing all of the forces acting on the particles. (b) (c) Derive the equation of motion of this mechanical system in the form (d) Show that Determine the changes in spring forces from their equilibrium values in terms of the above variables and parameters. 0--0) = A where A is a matrix that you should specify explicitly. (9) > is an eigenvector of A by calculating the corresponding eigenvalue. What is the physical significance of this eigenvector in terms of the motion of the particles?

Elements Of Electromagnetics

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ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

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Phase diagrams or equilibrium phase diagrams are the graphical representations that show the phase transformations of the constituting elements with respect to the pressure, temperature, and compositions. The diagrams show the conditions that limit the different phases, namely solid, liquid, and gas. The diagram can also be utilized to verify the relations between the state variables. Most of the phase diagrams are temperature-pressure phase diagrams, with the variable parameter as the composition. Usually other variables such as solubility and pressure parameters are not indicated in the phase diagram. For a one-component system (unary phase diagram), the phase diagram is generally a function of temperature and pressure as the state variables, or temperature and volume. However, for a binary system (two-phase) having two components, the phase diagram is a function of temperature and composition as the state variables only. Coexisting of the different phases of the constituting elements is indicated in the phase diagram. A ternary phase diagram can be characterized by consisting of three phases in equilibrium.

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