Please solve all 3 problems on MATLAB and provide code and output.

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ME 350 Project 2

Posted February 26, 2019, Due March 5 2019

Higher level tools for roots of f(x) = 0 problems

Q. 1. The velocity of a falling parachutist is given by the following formula:

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݃݉

ܿ ൬1െ݁ିቀ

ቁ௧൰

Where g=9.81 m/s2

, the drag coefficient c=13.5 kg/s.

Compute the mass (m) of the parachutist so that v = 40.7 m/s when t = 12.8 s

Hint. This is an f(x) = 0 type problem, with corresponding variables v(m) = 0

(a) Rearrange the terms and write a simple m-file, then use this file to use a

regular plot command to locate the approximate root. Submit the m-file and

the corresponding plot. Here you will need to experiment with a range of

values of ‘m’ .

(b) Modify the m-file from (a) into a function file that can be used with the

fzero function to compute a refined root. Submit your function and the

solution it displays in the MATLAB command window.

Q.2. Consider the square cross section beam anchored on one end and supporting a

500 lb weight. Other details are shown in figure below. Ref. Musto ch. 6. This

exercise will help you review methods for solving f(x) = 0 problems.

According to strength of materials principles, the maximum bending stress

experienced by the cantilever beam shown above is:

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σ = 335,000/x3

+ 92,600/x (1)

This reflects the suspended weight, the specific weight (weight per unit length),

and the length of the beam. From materials science, the maximum allowable stress

(i.e. limiting stress before the onset of yielding) for the beam material is 17,750

psi. We now need to determine the minimum dimension ‘x’ in inches that will satisfy

this stress constraint. Solve this problem using four different techniques as

itemized below.

Hint: The solution approaches involve solving a problem of the form: f(x) = 0

a) Rewrite equation 1 in polynomial form in descending coefficients of the

powers of x then use the built-in MATLAB function roots to compute the

representative solution.

b) Rewrite equation 1 into a MATLAB function file (beam2b.m) which can be

used by MATLAB’s fzero function. Use the fzero function to compute the

solution for this problem, using an initial guess of 43.

c) Rationalize the denominators of beam2b.m to generate a more standard

polynomial form and name it beam2c.m . Use fplot to generate a plot of

the function in the range -5≤x≤15; insert grid lines to spot the

approximate location of the real root. Use the fzero function to compute

the solution for this problem using and initial guess of -125. Submit the

function file, the plot and the solution computed using fzero.

d) Refer to the plot from (c). Use the bisectrr function to compute a root for

this function in the interval -5≤x≤5.

i. Comment on the outcome of the attempt in (d).

ii. Select an alternative range that would yield the root for this

function. What is the solution?

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Q.3. In designing a spherical tank whose schematic is shown below, you need to

determine the height of water in the tank if the radius R of the tank is 2.7 m, and

the volume V is 37.8 m3

. The equation for computing V as a function of R and h is:

ܸ ൌ ߨ݄ଶ ඌ3ܴ െ ݄

3 ඐ

Treat this is an f(h) = 0 problem to compute the height of water in the tank for

the given values of V and R

(a) Write a MATLAB function file that computes the volume the fluid in the

spherical tank as a function h; and generates a plot of V versus h.

(b) From your plot determine approximate of root for this function.

(c) Use fzero with the aid of the approximate root from (b) to compute the

desired root accurately.

Submit:

The MATLAB function file you used for (a) above

The plot generated with axis labels and grid lines showing

The approximate root from (b) above

The accurate root computed using fzero that is echoed to the

command window

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