Two Rings Attracting (Part 2)

Classical Mechanics Level pending

There are two circular rings, each with 1 1 unit of mass per unit length.

Ring 1 1 has a radius of 1 1 and center ( x , y , z ) = ( 0 , 0 , 0 ) (x,y,z) = (0,0,0) . The vector ( 1 , − 2 , 3 ) (1,-2,3) is normal to the plane of ring 1.
Ring 2 2 has a radius of 2 2 and center ( x , y , z ) = ( 1 , 2 , 3 ) (x,y,z) = (1,2,3) . The vector ( − 5 , 4 , 1 ) (-5,4,1) is normal to the plane of ring 2.

What is the magnitude of the gravitational force exerted by one ring on the other?

Details and Assumptions:
1) Universal gravitational constant G = 1 G = 1 , for simplicity


The answer is 7.26.

Steven Chase by Steven Chase , 37, USA

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1 solution

Karan Chatrath
Nov 8, 2019

Attached simulation code (MATLAB) below. I will include an explanation involving analytical expressions maybe later, as well. 

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clear all
clc

% Unit vectors characterising the larger ring:
R1 = 2;
k1 = [-5;4;1];
k1_hat = k1/norm(k1);
i1 = [1;1;7] - [1;2;3];
i1_hat = i1/norm(i1);
j1_hat = cross(k1_hat,i1_hat);

% Unit vectors characterising the smaller ring:
R2 = 1;
k2 = [1;-2;3];
k2_hat = k2/norm(k2);
i2 = [1;1;1/3] - [0;0;0];
i2_hat = i2/norm(i2);
j2_hat = cross(k2_hat,i2_hat);

Fx = 0;
Fy = 0;
Fz = 0;
dx = pi/5000;
dy = pi/5000;

for x = 0:dx:2*pi
    for y = 0:dy:2*pi

        % Parameterisation of the larger ring:
        r1 = [1;2;3] + R1*cos(x)*i1_hat + R1*sin(x)*j1_hat;

        % Parameterisation of the smaller ring:
        r2 = [0;0;0] + R2*cos(y)*i2_hat + R2*sin(y)*j2_hat;

        % Newton's Law of Gravitation:
        rd = r2 - r1;
        rm = sqrt(rd(1)^2 + rd(2)^2 + rd(3)^2);
        dF = (((R1*R2)/rm^3)*rd);

        % Extracting x,y,and z components
        dFx = dF(1);
        dFy = dF(2);
        dFz = dF(3);

        % Solving Double Integrals
        Fx  = Fx + dFx*dx*dy;
        Fy  = Fy + dFy*dx*dy;
        Fz  = Fz + dFz*dx*dy;
    end
end

Fr = norm([Fx;Fy;Fz])

% Fr = 7.262608239158482

1 Helpful 1 Interesting 1 Brilliant 0 Confused

@Karan chatrath Sir please expain these problems using analytical expressions. Please

A Former Brilliant Member - 1 year, 3 months ago

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