TI-84 Evo BASIC: Hydraulic Radius and Chezy-Manning (Circular Pipes)

TI-84 Evo BASIC: Hydraulic Radius and Chezy-Manning (Circular Pipes)

Introduction

The program, HYDRAD makes three calculations concerning the flow of water through a sloped pipe:

* Wetted perimeter of the pipe. This is the arc length of the pipe that is touched by the flowing water. (U.S. units: feet)

* Hydraulic radius, which is mathematically defined as the cross area of the water flow (not necessarily the entire cross area of a pipe) divided by the wetter perimeter. (U.S. units: feet)

* Flow rate of the water through the pipe. (U.S. units: cubic feet per second)

The flow rate depends on the type of material the water flows through.

The general equation is:

Q = α ÷ n * A * Y^(2/3) * √S

where:

α = coefficient which varies depending on what type of units are used. If US units, specifically feet, are used, then α = 1.49 (ft^(1/3)/s). If SI units, specifically meters, are used, then α = 1.00 (m^(1/3)/s). (Unusual powers of units.)

n = Manning’s Roughness Coefficient which is dependent on the material the water flows through. Specific values for certain materials include:

Concrete: n = 0.013

Cast Iron: n = 0.014

Steel: n = 0.010

(Lindeburg)

The coefficient is unit-less.

A = the flow area, which is the total cross-section are which water can flow through. In the case of the pipe, A = π * r^2.

Y = the hydraulic radius, or the “radius” of the wet area that the water touches. In general, the hydraulic radius is the wetted area divided by the wetted perimeter.

In the case of the pipe, the area of the wetted segment is calculated as R = r^2 * (Θ – sin Θ) ÷ 2, which Θ is the angle of the wet pipe in radians. The wetted perimeter is Wp = r * Θ. (see diagram below)

S = slope of the pipe, which should not be 0. In this case, downward slope is considered to be positive. (feet/feet or meters/meters)

TI-84 Evo BASIC Program: HYDRAD (US Units)

Code:

ClrHome

Disp “CHEZY-MANNING”, “PART-FILLED PIPE”,

“RADIANS SET”

Radian

Disp “LENGTHS IN FEET”

Input “PIPE RADIUS? “, R

Input “HEIGHT OF H20? “, H

Input “SLOPE? “, S

Menu(“SELECT PIPE MATERIAL”, “CONCRETE”, 1,

“CAST IRON”, 2, “STEEL”, 3, “OTHER N VALUE”, 4)

Lbl 1: .013 → N : Goto 0

Lbl 2: .014 → N : Goto 0

Lbl 3: .01 → N : Goto 0

Lbl 4: Input “N? “, N : Goto 0

Lbl 0

2 * cos^-1(1 – H / R) → Θ

R * Θ → W

R² * (Θ – sin(Θ)) / 2 → A

A / W → Y

(1.49 / N) * (π * R²) * Y^(2/3) * √(S) → Q

Disp “WETTED PERIMETER:”, W, “HYDRAULIC RADIUS”, Y,

“FLOW RATE:”, Q

Examples

Example 1:

pipe radius: r = 2.4 feet

water height: h = 0.9 feet

slope: 0.13

concrete pipe

wetted perimeter: 4.299191011 feet

hydraulic radius: 0.5463307184 feet

flow rate: 499.7552462 ft³/s

Example 2:

pipe radius: r = 1.5 feet

water height: h = 0.02 feet

slope: 0.4

custom n: n = 0.2

wetted perimeter: 0.4904439191 feet

hydraulic radius: 0.0132918217 feet

flow rate: 5.90983011 ft³/s

Example 3:

pipe radius: r = 7.5 feet

water height: h = 3 feet

slope: 1.01

steel pipe

wetted perimeter: 13.90942827 feet

hydraulic radius: 1.808870611 feet

flow rate: 39284.78283 ft³/s

Sources

EngineerExcel. “Chezy Equation: Application In Open Channel Flow Calculations” 2026. Retrieved July 7, 2026. https://engineerexcel.com/chezy-equation/

Lindeburg, Michael R, PE. Quick Reference for the Civil Engineering PE Exam. Professional Publications, Inc. Belmont, CA. 1999. ISBN 1-888577-47-9. pg. 19.

National Weather Service. “Normal Depth Calculator” Weather.gov > Alaska-Pacific RFC > Normal Depth Calculator. Anchorage, AK. Retrieved 5, 2026. https://www.weather.gov/aprfc/normaldepthcalc

Wikipedia. “Chézy formula” Edited March 12, 2026. Retrieved July 5, 2026. https://en.wikipedia.org/wiki/Ch%C3%A9zy_formula

Happy Halloween, everyone! May the spooky season bring fun, laughs, candy, and good times,

Eddie

All original content copyright, © 2011-2026. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Python: Drawing a Ring of Circles

Python: Drawing a Ring of Circles

The goal: Draw a number of circles that revolve around the origin (0,0). The center of each circle is rotated counterclockwise. The angle of the rotation is determined by:

a = 360° ÷ n = (2 * π rad) ÷ n

Since the center of each circle is set at (0, 0), the center of each circle is located at (r * cos a, r * sin a) where r is the radius.

Python: Using Turtle (Numworks)

( radius = 50 pixels, n = 8 circles)

The script circring.py imports the Turtle module. The circle command draws a circle starting at the bottom of the circle going counter-clockwise. The bottom of the circle is located at the point (x, y – r) where (x, y) is the center and r is the radius.

Code:

from math import *

from turtle import *

# 2026-07-19 EWS

# turtle version

r=eval(input("radius? "))

n=int(input("# steps? "))

speed(7)

for i in range(n):

  # set angle

  a=i*2*pi/n

  # turtle circle starts at

  # bottom center point

  x=r*cos(a)

  y=r*(sin(a)-1)

  # move to starting point

  penup()

  goto(x,y)

  # draw circle

  pendown()

  circle(r)

  

# make turtle invisible

hideturtle()

Python: Using TI-Draw (TI-84 Evo)

( radius = 50 pixels, n = 8 circles)

The script CIRRNG84.py uses the TI Draw module. The TI-84 Evo (and I think the TI nSpire, not 100% sure) lacks the Turtle module and instead employs a calculator-specific drawing module. The draw_circle command calls for both the center point and the radius. No adjustments are necessary.

Code:

from ti_draw import *

from math import *

# 2026-07-19 EWS

# TI version non turtle

set_window(-160,160,-105,105)

print("grid -160,160,-105,105")

r=eval(input("radius? "))

n=int(input("# steps? "))

clear()

for i in range(n):

  # set angle

  a=i*2*pi/n

  # determine center

  x=r*cos(a)

  y=r*sin(a)

  # draw circle

  draw_circle(x,y,r)

# show picture

show_draw()

Source

Chartier, Tim and Langville, Amy. Nonstandard Notebook: Mathematically Rule Pages for Unruly Thoughts The University of Chicago Press: Chicago and London. 2024 ISBN-13 978-0-226-83090-2 <pg. 61-63>

If you get a chance, please get this book. It’s a journal of interesting curves involving polynomials, circles, lines, and waves. The authors uses these curves to illustrate what ruled paper would look like when it doesn’t consist of all straight lines. The concept of music, earthquakes, and heartbeats are illustrated, including equations for clovers, mandalas, and hearts. This book is an inspiration for me.

Eddie

All original content copyright, © 2011-2026. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

HP 67 Collection: October 2026

HP 67 Collection: October 2026

All examples are done in FIX 4. (DSP 4)

Earth's Gravity at a Specific Latitude

The true earth gravity force depends on several factors, such as latitude and altitude. This estimation takes the latitude (north/south), in degrees-minutes-seconds, and latitude, in kilometers, into account.

The constant 9.80665 m/s^2 is an accepted average, the true force varies.

g_s = 9.806 – 0.026 * cos(2 * φ)

g_h = g_s * (6378.1 / (6378.1 + h))

HP 67 Code:

LBL A: 001 31, 25, 11

DEG: 002 35, 41

H←: 003 31, 74

2 : 004 02

× : 005 71

COS: 006 31, 63

. : 007 83

0 : 008 00

2 : 009 02

6 : 010 06

× : 011 71

9 : 012 09

. : 013 83

8 : 014 08

0 : 015 00

6 : 016 06

x<>y: 017 35, 52

- : 018 51

x<>y: 019 35, 52

6 : 020 06

3 : 021 03

7 : 022 07

8 : 023 08

. : 024 83

1 : 025 01

+ : 026 61

LST x: 027 35, 82

x<>y: 028 35, 52

÷ : 029 81

× : 030 71

RTN : 031 35, 22

Inputs:

Y: h

X: φ

Examples:

Inputs: h = 0, φ = 2°33’: g ≈ 9.7801

Inputs: h = 0.762, φ = 52°17’: g ≈ 9.8114

Source 

Grainger Engineering Office of Marketing and Communications. (answer written by Rebecca H.) (2016, November 21). “How gravitational force varies at different locations on Earth.” Illinois. https://van.physics.illinois.edu/ask/listing/64061. Retrieved March 10, 2026.

Triangulation

With angles α and ß:

base = dist * (1/tan α + 1/tan ß)

dist = base ÷ (1/tan α + 1/tan ß)

Labels:

LBL A: Enter angles α and ß in the current angle mode. Returns 1/tan α + 1/tan ß.

LBL B: Base. When a number is entered prior to pressing [ B ], the number is stored as a base. Otherwise, the program solves for a base.

LBL C: Distance (of a line that bisects the base). When a number is entered prior to pressing [ C ], the number is stored as a base. Otherwise, the program solves for a base.

HP 67 Code:

LBL A: 001 31, 25, 11 (enter the angles)

TAN: 002 31, 64

1/x: 003 35, 62

x<>y: 004 35, 52

TAN: 005 31, 64

1/x: 006 35, 62

+: 007 61

STO 5: 008 33, 05

RTN: 009 35, 22

LBL B: 010 31, 25, 12 (base)

STO 3: 011 33, 03

F? 3: 012 35, 71, 03

RTN: 013 35, 22

RCL 4: 014 34, 04

RCL 5: 015 34, 05

× : 016 71

STO 3: 017 33, 03

RTN: 018 35, 22

LBL C: 019 31, 25, 13 (distance)

STO 4: 020 33, 04

F? 3: 021 35, 71, 03

RTN: 022 35, 22

RCL 3: 023 34, 03

RCL 5: 024 34, 05

÷ : 025 81

STO 4: 026 33, 04

RTN: 027 35, 22

Examples:

α = 30°, ß = 65°

Set degree mode. 30 [ ENTER ] 65 [ A ]. Result: 2.1984

Distance = 24, solve for base. 24 [ C ] [ B ]. Result: 52.7606

Base = 24, solve for distance. 24 [ B ] [ C ]. Result: 10.9172

Source:

“Triangulation (surveying)” Wikipedia. https://en.wikipedia.org/wiki/Triangulation_(surveying) (last edited October 24, 2025). Retried March 17, 2026

Time of Runoff on Turf

t = 2.5 * (L / (S * I^2))^(1/3)

t: time of runoff (rain off) in minutes

2.5: turf constant

L: distance from remote area in feet

S: slope of the terrain

I: rain intensity in inches per hour

Input Stack:

Z: L

Y: S

X: I

Output Stack:

X: t

HP 67 Code:

LBL A: 001 31, 25, 11

x² : 002 32, 54

× : 003 71

÷ : 004 81

3 : 005 03

1/x: 006 35, 62

y^x: 007 35, 63

2 : 008 02

. : 009 83

5 : 010 05

× : 011 71

RTN: 012 35, 22

Examples:

Inputs: Z: 900 ft, Y: 0.11, X: 4 in/hr. Result: 19.9917 min

Inputs: Z: 650 ft, Y: 0.33, X: 0.06 in/hr. Result: 204.4744 min

Source

Texas Instruments. Slide Rule Calculator SR-50A Manual. 1975. Manual from datamath.org: http://www.datamath.net/Manuals/SR-50A_US.pdf pg. 59

Beamwidth of a Radio Telescope

This program calculates the beam width of a radio telescope in degree-minutes-seconds.

Θ = 67625 ÷ (D * f)

Θ = beam width

D = the diameter of a circular telescope in feet

f = observing frequency in MHz (megahertz)

Inputs:

Y: D

X: f

Outputs:

X: Θ

HP 67 Code:

LBL B: 001 31, 25, 12

× : 002 71

1/x : 003 35, 62

6 : 004 06

7 : 005 07

6 : 006 06

2 : 007 02

5 : 008 05

× : 009 71

→H.MS: 10 32, 74

RTN: 11 35, 22

Examples:

Inputs: X: 1980 MHz, Y: 106 ft. Output: Θ: 0.1920 (19’20”)

Inputs: X: 1665 MHz, Y: 84 ft. Output: Θ: 0.2901 (29’01”)

Source

Ball, John A. Algorithms for RPN Calculators John Wiley & Sons: New York, NY. 1978. ISBN 0-471-03078-8. pg. 259

Fan Laws

p2 = (rpm2 ÷ rpm1)^n * p1

(p1 and p2 stand for various parameters, see the input syntax below)

Inputs:

1^st law: (cfm: cubic feet per minute, rpm: revolutions per minute)

Calculate cfm2:

T: cfm1

Z: rpm2

Y: rpm1

X: 1

2^nd Law: (sp: static pressure, rpm: revolutions per minute)

Calculate sp2:

T: sp1

Z: rpm2

Y: rpm1

X: 2

3^rd Law: (hp: horsepower, rpm: revolutions per minute)

Calculate hp2:

T: hp1

Z: rpm2

Y: rpm1

X: 3

HP 67 Code:

LBL B: 001 31, 25, 12

R↓ : 002 35, 53

÷ : 003 81

R↑ : 004 35, 54

y^x: 005 35, 63

× : 006 71

RTN: 007 35, 22

Examples

1^st Law: T: cfm1 = 2000, Z: rpm2 = 1100, Y: rpm1 = 950, X: n = 1

Result: cfm2 ≈ 2315.7895

2^nd Law: T: sp1 = 0.5, Z: rpm2 = 1100, Y: rpm1 = 950, X: n = 2

Result: sp2 ≈ 0.6704

3^rd Law: T: hp1 = 1.5, Z: rpm2 = 1100, Y: rpm1 = 950, X: n = 3

Result: hp2 ≈ 2.3286

Source

Gladstone, John. Air Conditioning Testing and Balancing: A Field Practice Manual A Van Nostrand Reinhhold Book. August 1976. pp. 24-32

More HP 67 goodness to come in November,

Eddie

All original content copyright, © 2011-2026. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

HP 42S/DM42/Free 42: Two Functions Programmed Three Different Ways

HP 42S/DM42/Free 42: Two Functions Programmed Three Different Ways

Introduction

In 1979, Electro Scientific Industries (esi) released the (rare) ESI Calcumeter 4100 which is a combination multimeter and RPN calculator. I’m going to focus on the calculator part.

The calculator is a standard RPN calculator with the following functions: x^2, √, 1/x, 10^x, e^x, LN, log, and y^x. Also present are the standard ↓ROLL (roll down) and x<>y (swap) functions. Then we have other unique function such as:

mx+b: Applies a linear regression model to a measurement, with m stored in memory and b is grabbed from the y-registered

dBV: Decibel measurement (volt): 20 * log(measurement ÷ memory) (measurement is probably in the X register)

%DEV: percent change of a measurement from the value stored in memory:

100 * (measurement – memory) ÷ memory

AVG: average of cumulative amount of measurements against the value stored in memory

xy/(x+y): calculates x * y ÷ (x + y)

xy/(x – y): calculates x * y ÷ (x – y)

Getting this multimeter combo was not inexpensive, it had a retail price of $399 in 1979.

Today’s blog will focus on two of these functions, particularly xy/(x+y) and xy/(x-y).

The Functions xy/(x+y) and xy/(x-y)

As I could gather from the calculator’s keyboard:

xy/(x+y): Multiply Over Plus*

MOP(x,y) = (x * y) ÷ (x + y) = 1/(1/y + 1/x)

xy/(x-y): Multiply Over Subtract*

MOP(x,y) = (x * y) ÷ (x - y) = 1/(1/y – 1/x)

* These are my names for these functions. I don’t have the manual. Here I am assuming that the stack is in the traditional order: X is the display, Y is the stack just above X, and the RPN works like any (HP) RPN calculator.

All programs are made using the Free42 app (42S/41C family).

Stage 1: The Long Way

This starts out by duplicating x and y first until we have the stack as follows: Y, X, Y, X.

00 { 19-Byte Prgm }

01▸LBL "MOP"

02 ENTER

03 ENTER

04 CLX

05 -

06 R↓

07 X<>Y

08 +

09 R↓

10 ×

11 R↑

12 ÷

13 RTN

14 END

00 { 19-Byte Prgm }

01▸LBL "MOS"

02 ENTER

03 ENTER

04 CLX

05 -

06 R↓

07 X<>Y

08 -

09 R↓

10 ×

11 R↑

12 ÷

13 RTN

14 END

Stage 2: What if we assume x and y are nonzero?

Let n = (x * y) ÷ (x ± y)

Then:

1/n = (x ± y) ÷ (x * y)

1/n = x ÷ (x * y) ± y ÷ (x * y)

1/n = 1/y ± 1/x

n = 1/(1/y ± 1/x)

00 { 14-Byte Prgm }

01▸LBL "MOAR"

02 1/X

03 X<>Y

04 1/X

05 +

06 1/X

07 RTN

08 END

00 { 15-Byte Prgm }

01▸LBL "MOSR"

02 X<>Y

03 1/X

04 X<>Y

05 1/X

06 -

07 1/X

08 RTN

09 .END.

Shorter.

Stage 3: Using Stack Storage Operations (41C/42S only)

The HP 41C and 42S families allow us to use stack levels (X, Y, Z, T, and L) as storage registers. Let's see if we can take advantage of this.

00 { 17-Byte Prgm }

01▸LBL "MOP42"

02 STO ST Z

03 X<>Y

04 STO× ST Z

05 +

06 ÷

07 RTN

08 END

00 { 17-Byte Prgm }

01▸LBL "MOS42"

02 STO ST Z

03 X<>Y

04 STO× ST Z

05 -

06 ÷

07 RTN

08 END

Sources

Electro Scientific Industries, Inc. “Calcumeter 4100” Ad/Pamphlet Micrometrics, Inc. Portland, OR. 1979

Sanden, Kees van der. “The ESI Calcumeter 4100” https://www.keesvandersanden.nl/calculators/esi4100.php Retrieved June 19, 2026.

I hope you enjoyed our deep dive. See you next time,

Eddie

All original content copyright, © 2011-2026. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.

Casio fx-50F/Radio Shack EC-4024 Program Collection: October 2026

Casio fx-50F/Radio Shack EC-4024 Program Collection: October 2026

Notes:

* The fx-50F/EC-4024 has a 29 step program space between two program spaces. Program steps can not be edited but can be cleared by [SHIFT] [ C ] (PCL) key sequence.

* Formula calls can not be included as a program step, but scientific constants can be. The values for the constants most likely are from the Japan Standards of 1978 (JISZ-8020-1978), so a little dated to 2026 standards.

* The Casio fx-10F is similar as fx-50F/EC-4024 except that the fx-10F does not have the scientific constants and has a slightly inclined keyboard.

* Mode changes can be included. SHIFT and Kin arithmetic calls are merged.

The programs presented today require inputs to be stored in registers 1-6 using [ SHIFT ] [ Kout ] (Kin) ((K)onstants In).

Angle Between 2 Lines (in Degrees)

Θ = arctan( abs((y – x) ÷ (1 + x * y)) )

abs(x) = √(x^2), key sequence: [ x² ] [ √ ]

x: slope of line 1 (Kin 1)

y: slope of line 2 (Kin 2)

Program Code (20 steps):

MODE 4 (degree)

(

(

Kout 2

-

Kout 1

)

÷

(

1

+

Kout 1

×

Kout 2

)

)

x²

√

TAN^-1

=

Maximum and Minimum of Two Numbers

The two numbers are stored in registers 1 and 2 (Kin 1, Kin 2).

max(x,y) = (x + y + abs(x – y)) ÷ 2 = min(x,y) + abs(x – y)

min(x,y) = (x + y - abs(x – y)) ÷ 2 = max(x,y) - abs(x – y)

Program Code (21 steps):

Kout 2

-

Kout 1

=

x²

√

Kin 3

(

Kout 1

+

Kout 2

+

Kout 3

)

÷

2

=

HLT (show maximum)

-

Kout 3

= (show minimum)

Terminal Velocity of a Spherical Object

vt = √(2 * g * mass ÷ (ρ * cd * exposed area of the object) (units: m/s)

≈ √(34.05656 * mass ÷ (π * radius^2))

The following constants are assumed:

g = 9.80665 m/s² (Earth’s gravitational constant)

ρ ≈ 1.225 kg/m³ (air pressure)

cd = 0.47 (coefficient drag of a sphere)

2 * g ÷ (ρ * cd) ≈ 34.06565 m^4/(s^2 kg)

Program Code (19 steps):

Kout 1

×

3

4

.

0

6

5

6

5

÷

(

π

×

Kout 2

x²

)

=

√

=

Ideal Thermal Voltage and Shockley Diode Equation

Thermal Voltage:

vt = k * T ÷ q * 10^3 (mV)

Shockley Diode Equation:

Id = Is * (exp(vD ÷ (n * vt)) – 1)

Ideal: n = 1

Inputs:

Is: scale current in milliamps (mA) Kin 1

Vd: diode voltage in millivolts (vA) Kin 2

T: temperature in degrees Kelvin (K) Kin 3

Outputs:

vt: Thermal Voltage (mA),

Id: diode current (mV)

Constant used (see comments on the scientific constants above):

k: Boltzmann Constant ([SHIFT] [ 8 ]): 1.380662 * 10^-23 J/K

(q) e: Electron Charge ([SHIFT] [ 4 ]): 1.6021892 * 10^-19 C

Program Code (21 steps):

Kout 3

×

k (SHIFT 8)

÷

e (SHIFT 4)

×

3

10^x

=

HLT (see vt: thermal voltage)

1/x

×

Kout 2

=

e^x

-

1

=

×

Kout 1

=

Source:

“Shockley Diode Equation” Wikipedia. Last Edited November 13, 2025.

https://en.wikipedia.org/wiki/Shockley_diode_equation

Retrieved June 19, 2026.

Eddie

All original content copyright, © 2011-2026. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.