Create a main MATLAB script that performs the tasks described below: (a)Use MATL

Question

Create a main MATLAB script that performs the tasks described below:

(a)Use MATLAB to compute the magnitude of the Earth’s gravitational acceleration, from the ground up to the altitude of the International Space Station (ISS), that is, 400 km.

(b) The electric field, due to a uniformly charged ring, at any point along the x axis (see Figure below) is given by:

Takinga=0.1m,plotinthesamefigure,Ex versusxforQ=2nCandQ=4nC.One curve must be represented by a solid black line and another curve must be represented by a dashed red line. Research on how to place labels on the horizontal and vertical axis, a legend for your curves, and present your plot with them. Pay attention to units when labeling your plots (Example: “Time [s]”).

Explanation / Answer

% XYZ2GRAV - computes the gravitational acceleration vector at a specified
% ECF location using the JGM2 gravitational ellipsoid only.
% Higher-order gravity terms (the "gravity anomaly") are
% ingnored. Only the pure ellipsoid is used.
%
% [gx,gy,gz] = XYZ2GRAV(x,y,z)
%
% x, y, z = Earth-Centered-Fixed (ECF) cartesian coordinates (meters)
% gx,gy,gz = components of gravitational acceleration in ECF (meters/sec^2)
%
% NOTES: (1) x,y,z may be scalars, vectors, or matrices but must have
% the same size and shape
% (2) gx,gy,gz will have the same size/shape as x,y,z
% (3) Only the purely ellipsoidal portion of the JGM-2 gravity
% model is used here
% (4) Points inside the earth will return gravity vectors, but
% will be higher in magnitude than physical reality since
% mass "above" the point is not removed from consideration;
% i.e., the ellipsoidal gravitational field surrounds a point at
% the center of the earth. The strength of the field increases
% as one gets closer to that central point.
% (5) If all inputs are zero, then NaNs are returned.
% (6) Source of initial formulas:
% www.colorado.edu/ASEN/asen3200/labs/ASEN3200_LabO3_2005.pdf
% (7) The gx & gy formulas were written in a more explicit form
% to prevent divergence when the x-coordinate is zero.
% (8) The calculation was accelerated by precomputing some values.
% (9) No warranty; use at your own risk.
% (10) Version 1.0 - Initial writing. Michael Kleder, August 2005
%
% % EXAMPLE:
% R=6378137;
% [x,y,z]=sphere(50);
% [x,y,z]=deal(x*R,y*R,z*R);
% [gx,gy,gz]=xyz2grav(x,y,z);
% gm = sqrt(gx.^2+gy.^2+gz.^2);
% figure
% surf(x,y,z,gm)
% axis equal
% colorbar
% title('Gravitational acceleration on a uniform sphere (m/s^2)')
% disp(['When standing at a pole, one experiences slightly MORE gravitational ' ...
% 'acceleration, but that is because one is closer to the center of ' ...
% 'the earth (smaller radius); however, the provided plot is of ' ...
% 'gravity on a uniform sphere -- equal distances -- so that gravity ' ...
% 'is greater along the equatorial plane.'])

function [gx,gy,gz] = xyz2grav(x,y,z)
J2 = .00108263;
mu = 3.986004418e14; % m^3/s^2
R = 6378137; % m
r = sqrt(x.^2+y.^2+z.^2);
sub1 = 1.5*J2.*(R./r).^2;
sub2 = 5.*z.^2./r.^2;
sub3 = -mu./r.^3;
sub4 = sub3.*(1-sub1.*(sub2-1));
gx = x .* sub4;
gy = y .* sub4;
gz = z .* sub3.*(1-sub1.*(sub2-3));
return

sum of infinite series

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