bethel-physics / hw_lineareqs

HW for Garcia ch. 4, linear systems of equations

hw_lineareqs

Contents

• Problem 1 (in-class)
• Problem23 (in-class?)

Problem 1 (in-class)

Consider the linear mass-spring system shown. All masses and natural (i.e. unstretched) spring lengths are equal, but the spring constants vary as indicated. The top spring is attached at y = 1.5 m and the +y-direction is upward.

• On paper, write expressions for the equilibrium conditions for each mass.
• Define your matrix A and the right-hand-side vector b for the linear system Ay = b describing the mass-spring system.
• Determine the equilibrium positions y for each mass using the MATLAB command, x = inv(A)b

Problem 2 (in-class?)

(From Garcia, #6 page 115) Using Kirchhoff's laws in circuit problems involves solving a set of simultaneous equations. Consider the simple circuit shown.

• Write a program that computes the currents, given the resistances and voltages as inputs.
• Have your program produce a graph of the power delivered to R₅ as a function of V₂
• Use this range of values for V₂: 0 -- 20 V
• Use R₁ = R₂ = 1 Ω, R₃ = R₄ = 2 Ω, R₅ = 5 Ω, V₁ = 2 V, V₃ = 5 V.

Important equations:

• Kirchhoff's loop rule: ΔV = 0 over any loop
• Kirchhoff's junction rule: net current flowing into a junction = net current flowing out of the junction
• Power delivered to a resistor: P = I²R