Location of the center of mass of a region

Description

Published by Marcel Pelletier, June 2023, in the MoHPC - General Software Library

This document shows how to use the integral function yx, in a program, to calculate the location of the center of mass of a region.

Center of Mass

Program C uses registers .0 , .1 and .2 . The user-defined functions, label A and label B, can be add at the end of the program memory.
The center of mass of a region (the centroid) is located at point (x, y), where
        b                     b                      b
x = 1/A ∫ x f(x) dx   y = 1/A1/2 (f(x))² dx   A = ∫ f(x) dx
        a                     a                      a

If the region lies between two curves y=f(x) and y=g(x), where f(x)≥g(x) then,
        b                              b                                  b
x = 1/A ∫ x [f(x) - g(x)] dx   y = 1/A1/2 [(f(x))² - (g(x))²] dx   A = ∫ [f(x) - g(x)] dx
        a                              a                                  a


Examples

Example 1
Find the centroid of the region bounced by the curves y=cos x, y=0, x=0∧x=π/2.

In PRGM mode:
Delete any existing program at label A.

f LBL A
COS
g RTN

In RUN mode:
g RAD
0 ENTER → 0.000
g π 2 ÷ → 1.5708
f C → 0.5708
x↔y → 0.3927

The centroid is located at the point (π/2-1, π/8).

Example 2
Find the centroid of the region bounded by the line y=x and the parabola y=x². The curves intersect at the point (0, 0) and (1, 1) → a=0 and b=1.

In PRGM mode:
Delete any existing program at label A.

f LBL A
g
g RTN
f LBL B
g
g RTN

In RUN mode:
g SF 1 (This the second case!)
0 ENTER
1 f C → 0.5000
x↔y → 0.4000

The centroid is located at the point (1/2, ²/5) or (0.5, 0.4).

Program Resources

Labels

Name Description Name Description
 A User-defined function A .3
 B User-defined function B .4
 C Center of Mass .5
.1 .6
.2

Storage Registers

Name Description
.0
.1
.2

Flags

Number Description
1

Program

Line Display Key Sequence Line Display Key Sequence Line Display Key Sequence
000 019 32 .2 GSB . 2 038 32 12 GSB B
001 42,21,13 f LBL C 020 45,20, .0 RCL × . 0 039 43 11 g
002 42,20, .5 f ∫xy . 5 021 43 32 g RTN 040 30
003 44 .1 STO . 1 022 42,21, .2 f LBL . 2 041 43 32 g RTN
004 33 R⬇ 023 45 .0 RCL . 0 042 42,21, .5 f LBL . 5
005 33 R⬇ 024 32 12 GSB B 043 44 .0 STO . 0
006 42,20, .1 f ∫xy . 1 025 30 044 32 11 GSB A
007 44 .2 STO . 2 026 43 32 g RTN 045 43, 6, 1 g F? 1
008 33 R⬇ 027 42,21, .3 f LBL . 3 046 32 .6 GSB . 6
009 33 R⬇ 028 44 .0 STO . 0 047 43 32 g RTN
010 42,20, .3 f ∫xy . 3 029 32 11 GSB A 048 42,21, .6 f LBL . 6
011 45,10, .1 RCL ÷ . 1 030 43 11 g 049 43 32 g RTN
012 45 .2 RCL . 2 031 43, 6, 1 g F? 1 050 32 12 GSB B
013 45,10, .1 RCL ÷ . 1 032 32 .4 GSB . 4 051 30
014 43 32 g RTN 033 2 2 052 43 32 g RTN
015 42,21, .1 f LBL . 1 034 10 ÷ 053 42,21,11 f LBL A
016 44 .0 STO . 0 035 43 32 g RTN 054 24 COS
017 32 11 GSB A 036 42,21, .4 f LBL . 4 055 43 32 g RTN
018 43, 6, 1 g F? 1 037 45 .0 RCL . 0