Question

A) Define a set of 5 4x4 matrices containing random floating-point numbers between 0 and 1. Use the appropriate routines in numpy.linalg and scipy.linalg to find the determinants of each matrix, in the case of numpy without using a loop. Make a quantitative comparison of the results between the two methods.

b)We want to write the above equations in matrix form as F = Kx - b. Write a function that takes the ki and Li values as inputs in 1-D arrays and returns the arrays corresponding to K and b.
c)Use the function from part b and the linalg functions to solve for the equilibrium positions of the masses (where Fi=0) for the cases
(i) k = [1 2 3 4], L = [1 1 1 1], Lw=10
(ii) k = [0 1 1 0], L = [2 2 1 1], Lw=4
d)Write the appropriate code needed to solve this system of equations using scipy.integrate.solve_ivp. (Hint: your state vector in this case should have 6 components, 3 for x_i and 3 for v_i. Your derivative function will need the vectors k and L as additional arguments). Compute the solutions using as initial values the equilibrium positions x_i for the 2 cases you computed in part 3, and v_i=0. Are the results as you might expect?

Answer 1

The student asked about computing determinants of random matrices, solving for equilibrium in a physical system, and utilizing scipy's integration methods to solve equations of motion, all within the realm of computational physics.

Step-by-step explanation:

The student's question involves defining a set of 5 4x4 matrices with random floating-point numbers between 0 and 1 and computing their determinants using routines in numpy.linalg and scipy.linalg, comparing the results quantitatively. Additionally, a function needs to be written to convert the given spring constants (ki) and natural lengths (Li) into matrix form and use it along with linear algebra methods to solve for equilibrium positions. The final part involves solving a system of equations with initial conditions using scipy.integrate.solve_ivp for two given cases. It is essential to draw free-body diagrams, list knowns and unknowns, and find the appropriate equations to solve for unknowns within the context of physics problems.