CS 1112 Project 2, part A due Mon, Mar. 8, 11pm EST

Rocket fuel gauge

A rocket’s liquid oxygen tank is formed as a cylinder of radius r with hemispherical end caps (also of radius r), as shown in the figure to the right. The tank has a height h, where h ≥ 2r (which would be the height of the caps alone). By measuring the difference in pressure between the top and bottom of the tank, engineers can infer the depth d of propellant in the tank. But in order to know how much longer they can fire the engines, they must determine the volume of propellant remaining.

Write a script to compute the volume of liquid in the tank at n different “fill depths” d and make a plot of volume versus depth. The n values of d should be equally spaced from 0 to h, inclusive; your script should prompt the user to input a value for n, which you may assume will be an integer greater than 2. Solicit user input for r and h in meters as well. If the user-provided value of h is less than or equal to 2r, set h to 2r and print a message to the Command Window showing the new value of h.

Your script tank should have the following “outline”:

% Solicit input for r, h, and n. Change h if necessary.

close all    % Close all figure windows
figure       % Start a figure window
hold on      % Keep all subsequent plot commands on same axes

% Your code to calculate the volume and plot volume versus depth

hold off     % Subsequent plot command is shown on fresh axes

Note these hints and additional specifications:

• Given n and h, you need to determine the different values of d. What is the “step size” from one d value to the next?
• Break down the problem; don’t try to do everything at once! Specifically, leave the plotting to the end. First develop the code to calculate the volume at the various depths (consider printing the results in a table to check your work); then do the graphics.
• The volume of a spherical cap of radius r filled to a height b (measured from the round end) is given by the formula $V_\text{cap}=\frac{\pi}{3}b^2 (3r - b)$ .
• The plot has volume on the y-axis. Draw one marker (e.g., an asterisk, a cross, …) for the volume at each depth; do not connect the markers with lines. Recall that the statement plot(x,y,'k*') draws a black asterisk at the position (x,y).
• Use three different marker colors for fill depths rising to each of the three segments of the tank: the bottom spherical cap, the top spherical cap, and the segment in between.
• Label the axes and add a title to the plot; don’t forget units! The commands to use are title(), xlabel(), and ylabel(), and they work the same way. For example, to label the x-axis with “fill depth [m]”, write the statement xlabel('fill depth [m]'). Check the Matlab documentation for further details if necessary.
• Before submitting, check your plot for consistency. Does the trend look continuous and monotonic? If not, use any deviations from expected behavior to help pinpoint your bug.

Polar 3D printer

In a 3D printer, plastic is extruded through a nozzle over a build surface. By moving the nozzle relative to the surface, material can deposited in an arbitrary pattern within a horizontal layer, and by stacking many such layers, a three-dimensional object is formed. In many 3D printers, the nozzle is able to move left and right along the x-axis (from a top-down view) while the build surface slides along the y-axis (with layers stacking up the z-axis).

However, in a polar 3D printer, the build surface is a circular turntable that spins about its center, while the nozzle is attached to an arm that pivots about a point just off the edge of the turntable (see figure). By making the length of the pivot arm equal to the radius of the turntable, the nozzle can be positioned over any point on the turntable by controlling a combination of spin angle φ (“phi”) and pivot angle θ (“theta”). Printing circular shapes is easy with such a setup (the turntable just spins under a fixed nozzle), but printing straight lines evenly is more difficult.

Our goal is to print a straight line from the center of the build surface to its edge, represented by the dotted line in the figure. Initially, both φ and θ are 0°; then, as the turntable spins (increasing φ), we must pivot the arm (increasing θ) at a different rate so that the nozzle (denoted by P) stays on the line. While the required pivot can be computed analytically, we will instead take an iterative approach inspired by feedback control: starting with the nozzle over the center (θ = 0°), increase θ by a fixed amount until the angle that the nozzle makes with respect to the horizontal line in the figure is equal to the turntable angle φ. More specifically, we want to increase θ until the difference between φ and the angle to P changes sign; that final value of θ is our solution. Along the way, we should plot this difference vs. θ to visualize our approach.

Download the file polarPrinter.m from the Projects page. Read and run it. Since the program is incomplete, the output includes only a figure window with axis labels and the given values of r and φ printed to the Command Window. You will complete the program to determine and print to the Command Window the value of θ that would place the nozzle on the target line, as well as its distance s along that line (you will use this to evaluate the evenness of the line in Project 3). You must use the approach described above (though if you can work out the analytical solution, you may use it to check your work).

Additional specifications, considerations, and hints

• Start by choosing a coordinate system. Placing the origin at the center of the turntable will be easiest, but you may try placing it at the pivot point if you like.
• Solve for the x and y coordinates of P in terms of θ. Then relate their ratio to the angle that P makes with respect to the x-axis. Matlab’s built-in function to compute the inverse tangent is called atan(), but when computing the inverse tangent of a ratio y/x, atan2(y,x) is preferred. Remember that Matlab’s trigonometric functions work in radians.
• Choose an appropriate step size for incrementing θ. Some amount of error is expected, so use your best judgment—and experimentation—to come up with a reasonable value. The graph that your program produces will help you see when a step size is too big.
• A hint for detecting a sign change: If ab ≤ 0, then either a and b have different signs, or a and/or b is zero.
• Add an fprintf() statement to print your solution for θ and s to the Command Window. Print their values, in units of degrees and centimeters, to two decimal places.
• To add a point to your graph, use the plot() function. For example, plot(x,y,'b.') plots a blue dot. You may use any color and marker you like.
• Add a title to the figure displaying the values of φ and θ. Review Project 1 if you need a reminder on how to use the title() and sprintf() functions.
• The provided code specifies a set of values for r and φ, but your program should still work if the values are changed! You may assume that r > 0 and that 0° ≤ φ ≤ 30°. Test your program with other reasonable values, but change them back to the original ones before submitting.