Mathematical Methods for Engineers 2

Assignment Task

A. (a) Assuming the ladder is initially at rest (w(0) = 0), solve for the seven output values in the matrix equation if the ladder is positioned at an angle of θ₀ = π/6 to the wall. Write your answers in your report (to 4 decimal places).

(b) Set up an array of 0 values, θ₀  0< θ < π/2. Use a for loop to find the corresponding three accelerations ax, ay, a θ₀ at all of these initial values by repeatedly solving the matrix equation, and so create three arrays a x, a y, a theta of initial accelerations depending upon θ₀.

(c) Create a plot (Figure 1) (with legend, labels on the axes, title, etc) that has θ0 on the horizontal axis and the three initial accelerations on the vertical axis (make ax blue, ay red and a θ₀ black). You will see that the three curves all pass through the same point on the horizontal axis, corresponding to a critical value of θ₀.

Question 

(a) Why are the sections to the left of the critical value of θ₀ meaningless? What should they be?

(b) Using the first, second, third and fifth of the linear equations (in the matrix equation) prove that if all three accelerations are initially zero, then tan  θ₀= 2µk /1- µ2 k .

(c) Evaluate the critical value of θ₀(when the ladder starts from rest) (2 decimal places), giving your answer in radians. Write your answer in your report.

B. 2 (a) Use the ode45 solver to solve for the motion of the falling ladder, assuming that the initial angle of the ladder is θ₀ = π/6 (which is more than the critical value).

(b) The ladder will slip down the wall, with the motion described by the di↵erential equations, until it hits the floor. The problem is that you do not know the time tend when this occurs. You can assume it is at most 2.5 s. Search (by varying tend) until you find tend correct to 4 decimal places (write this number in your report), and then produce two separate graphs of  θ against t, and of ! against t for 0 < t>

Question

(a) Give a physical interpretation of the initial conditions ( θ(0) =  θ₀ = 0 and !(0) = 0).

(b) Once again, the equation of motion can be rewritten as two first order equations. In your report, rewrite equation (2) as two first order equations.

C. (a) Use ode45 to produce a graph of θ(t) for the first 20 seconds (Figure 4). You will need a separate function to enter the equations to describe a swinging rod.

(b) Use max to estimate the maximum angle the rod reaches on its first swing, and min to find the angle the rod reaches on its first return. Give these two angles in radians, to 4 decimal places (write these two numbers in your report).

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