Moment Calculation (5-22-2020)

Classical Mechanics Level pending

A solid uniform spherical ball of mass 1 1 and radius 1 1 is centered on the origin. Consider the portion of the ball above the plane x + 3 y + 2 z = 1 x + 3y + 2z = 1 . What is the moment of inertia of this object about the z z axis?

Note: "Above" means greater z z


The answer is 0.135.

Steven Chase by Steven Chase , 37, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Karan Chatrath
May 22, 2020

Executing a script of code is a very straightforward way of solving this problem. I will try and update this solution with an analytical one later. 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
 
clear all
clc
% Numerical resoloution:
dx = 1e-3;
dy = 1e-3;
dz = 1e-3;

% Mass volume element computation: (Density X Elementary_Volume)
M  = 1;
R  = 1;
V  = ((4*pi)/3)*R^3;
dM = (M/V)*dx*dy*dz;

% Moment of inertia about Z initialisation:
I  = 0;

for x = -1:dx:1
    for y = -sqrt(1-x^2):dy:sqrt(1-x^2)
        for z = -sqrt(1-x^2-y^2):dz:sqrt(1-x^2-y^2)

          % Check if point lies above the plane:  
          if x+3*y+2*z>1  
              dI = dM*(x^2 + y^2);

              % Numerical integration:
              I  = I + dI;
          end

        end
    end
end

Answer = I;
% Answer = 0.1353

1 Helpful 1 Interesting 1 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...