Numworks (Python): Determining Earth’s Acceleration of Gravity

Numworks (Python): Determining Earth’s Acceleration of Gravity

Introduction

Note: We will only be using SI units on this blog entry and program.

In general, the Earth’s acceleration constant, labeled with a lower case g, is conventionally defined to be 9.80665 m/s². Did you know that the gravitation acceleration on this planet actually varies (albeit slightly) depending on factors? For example, the gravity’s acceleration in Los Angeles, California, is about 9.796 m/s² (see Wikipedia article in the Source section).

On today’s blog, we will estimate the gravity of Earth based with two parameters: the latitude (North/South position on Earth’s grid) and elevation (in meters). This will give us a more accurate estimated on gravity, which in turn will give other physics calculations more accuracy.

For this program, I’m using a mix of formulas to estimate Earth’s gravitational acceleration:

Let Φ = latitude in radian measure:

Φ = (degrees + minutes ÷ 60 + seconds ÷ 3600) * π ÷ 180

For this calculation, consider the absolute value of the latitude. Gravity acceleration is the a given the northern latitude north and the corresponding southern latitude. (Example: gravity is about 9.79325 m/s² at 30° north and 30° south).

Gravity at the surface (0 m) is calculated as:

gs = (a * ga * (cos Φ)^2 + b * gb * (sin Φ)^2) ÷ √((a * cos(Φ))^2 + (b * sin(Φ))^2)

[Bilsin]

Where:

Φ = latitude in radians (see formula above)

a = 6378137 m (Earth’s radius at the equator)

ga = 9.70803253359 m/s² (gravity at the equator)

b = 6356752.3142 m (Earth’s radius at the poles)

gb = 9.8321849378 m/s² (gravity at the poles)

[Bilsin, Table 2]

Gravity at the selected elevation is estimated at:

ge = (100 * gs – 0.3086 * h ÷ 1000) ÷ 100

[Glover]

Where:

h = height in meters (m)

Note: The table on the Desk Ref list gravity as cm/s² and the original formula called for the height in kilometers (km).

To convert feet to meters, multiply by 0.3048.

Numworks Program: earthg.py

https://my.numworks.com/python/ews31415/earthg

Code:

from math import *

# Numworks, EWS, January 2026

# constants

# equator: radius, gravity

a=6378137

ga=9.7803253359

# poles: radius, gravity

b=6356721.3142

gb=9.8321849378

print("Latitude:","\nNorth or South")

d=eval(input("Decimal? "))

m=eval(input("Minutes? "))

s=eval(input("Seconds? "))

phi=radians(d+m/60+s/3600)

print("Height (m)?")

h=eval(input("? "))

gs=a*ga*cos(phi)**2+b*gb*sin(phi)**2

gs/=sqrt((a*cos(phi))**2+(b*sin(phi))**2)

print("Gravity on the surface:")

print(str(gs)," m/s**2")

print("At altitude: ",str(h)," m")

ge=(100*gs-0.3086*h/1000)/100

print(str(ge)," m/s**2")

Examples

Latitude	Elevation	g_surface	g_elevation
30°0’0”	0	9.793247191840559	9.793247191840559
40°0’0”	304.8	9.801696762579571	9.800756149779572
45°0’0”	762	9.806197665936962	9.803846133936961
12°5’13”	1219.2	9.782589588493437	9.778827137293437
34°26’44”	500	9.796866770553882	9.795323770553882

g_surface: gravity at surface level (0 m)

g_elevation: gravity at the selected elevation

Sources

Bislin, Walter. “Earth Gravity Calculator” September 1, 2018. https://walter.bislins.ch/bloge/index.asp?page=Earth+Gravity+Calculator Retrieved January 15, 2026.

Glover, Thomas J. and Richard A. Young Desk Ref Sequoia Publishing, Inc. Anchorage, AK. 4^th Edition, 2022. Soft Cover ISBN 978-1-885071-60-6. pg. 587

“Gravity of Earth” Wikipedia. Last edited January 31, 2026. https://en.wikipedia.org/wiki/Gravity_of_Earth Retrieved February 1, 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.

The New TI-84 Evo

The New TI-84 Evo

Some Notes on the TI-84 Evo

The TI-84 Evo was released on April 28, 2026 in the United States. It is an update of the TI-84 CE Python and in fact, replacing the TI-84 CE Python. As of right now, the TI graphing calculator the TI-83/84 family consists of:

TI-83 Plus: This calculator model was first introduced in 1999 and is run by four AAA batteries with a CR1620 backup.

TI-84 Plus: This is the base TI-84 model with a monochrome screen, originally first arrived in 2004. The last firmware, 2.55, adds math print (textbook print).

TI-84 Plus CE: This is a color version of the TI-84 Plus and has a rechargeable battery.

TI-84 Evo: This is next version of a TI-84 Plus that has Python.

Quick Facts and What the TI-84 Evo Can Do

Model: TI-84 Evo

Company: Texas Instruments

Type: Graphing

Programming Language: TI-Basic, Python

Power: Rechargeable Battery, powered by as USB C cord

Case: Slide case

Memory: 7 memory registers: A, B, C, D, X, Y, M (M has memory addition and subtraction)

Years in Production: April 28, 2026 - present

Display: 320 x 240 pixels, 2.8” diagonal screen

Colors: White, Pink, Mint Green, Raspberry, Silver, Teal, Lavender. I have a white one.

Retail Price: $160 (US dollars). There is a four year license to an online emulator included.

For reference, the TI-84 Plus CE Python, the calculator the TI-84 Evo replaces, was in production from July 27, 2021 to April 27, 2026.

At this point we all know, more or less, the main features of the TI-84 Plus, as they are present with the current TI-84 Evo (not an all inclusive list):

* Graph up to 10 functions, six parametric functions, six polar functions, and three recursive functions

* Lists can have up to 999 elements can be used. Lists can have real and complex numbers. There are many functions associated with lists, such as sorting, finding the arithmetic average (mean), the sum of the elements, applying lists in statistical analysis, and generating lists from defining a sequence. There are six lists that can be accessed from the keyboard and additional lists with custom names can be created.

* Complex numbers including arithmetic, conjugate, conversion between rectangular (a+bi) and polar forms (re^(iΘ)), square, square root, cube, and cube root of complex numbers.

* 10 matrices, [ A ] through [ J ], including the transpose, inverse, determinant, row operations, and generating identity matrices.

* Many statistical regressions, distributions (normal, student, Chi-squared, F, binomial, Poisson), and ANOVA.

* Drawing tools

* Two programming languages: the classic TI-Basic and Python. Python modules include math, random, plotlib (TI version), time, specialized modules for the TI hub, TI rover, and import processing.

Two major selling points of the TI-84 Evo are:

* Distraction free mathematics with no smartphone interface. However, all calculators that are not downloaded to smartphones qualify to be “distraction free”. This is a response to curbing smartphone use in the classroom.

* The TI-84 Evo has an icon menu. The icon main menu replaces the app key. While the TI-84 Evo is on, the [ on ] key acts as toggle between the main menu and the last used app.

The apps included with the TI-84 Evo are:

1. Calculator (Home)

2. Y= Function Editor (same as pressing [Y=]

3. List Editor (same as pressing [ stat ], [ 1 ])

4. Mode Settings (same as pressing [ mode ])

5. Numeric Solver (same as pressing [math], C or [math], [ ↑ ], [ enter ])

6. Polynomial Root Finder

7. System Solver (linear systems)

8. Finance (time value of money solver and basic finance functions, including date functions covering years from 1980 to 2079).

9. Transformation Graphing

Inequality Graphing

Conics Graphing

Python

TI-Basic

Help (one page has a QR code for which leads to the TI-84 Evo online guide)

TI-84 Evo User Guide: https://education.ti.com/en/product-resources/eguides/eguide-84-evo

Some Differences Between the TI-84 Evo and the Previous TI-84 Plus CE Python

For this section, I refer the TI-84 Plus CE Python as the 84 Python and the TI-84 Evo as the 84 Evo. These are some of the changes observed. Despite these changes, at its core the TI-84 Evo is easy to pick up and learn, and if you are transferring from an older TI-82/83/84, the learning curve is at the most, minimal.

Keyboard Changes:

84 Python: smaller keys, 2nd and alpha functions printed above the keys 

84 Evo: big square keys, 2nd and alpha functions printed on the keys

84 Python  (TI-84 CE Python) Keyboard

84 Evo  (TI-84 Evo) Keyboard

From the 84 Python to the 84 Evo:

2nd of [del] key: ins becomes an icon | <> []

distr (distributions): moves from 2nd of [vars] to alpha of [stat]

[apps] key replaced with fraction template [ []/[] ]

2nd of [math] key: test becomes an icon =≤≠>

matrix: moves from 2nd of [x^-1] key to 2nd of [vars] key

[clear] key gets an undo clear 2nd function, labeled ⟲clear, can be used as "cut"/"copy" and "paste"

[x^-1] reciprocal key becomes [x^[]] power key template, while its 2nd function is the root template []√

Arithmetic keys move up a row: 

[ ^ ] becomes the [ ÷ ] key 

[ ÷ ] becomes the [ × ] key 

[ × ] becomes the [ - ] key 

[ - ] becomes the [ + ] key 

[ + ] becomes the [ <> ] key (utility/toggle key) Though the primary functions change the 2nd and alpha functions remain the same

[enter] key appears to lose the solve alpha function

2nd of [sto→] key: rcl gets lengthened to recall

[on] key gets a home icon

Battery:

84 Python: Lithium ion polymer 

84 Evo: Lithium ion

Power Connection:

84 Python: TI-specific cord with USB-A/USB-mini 

84 Evo: USB-C/USB-A (came in the box)

Colors of the keys – on the white keyboard:

84 Python: blue 2nd key, green alpha key, black math keys, gray arithmetic keys, white number keys, gray function and arrow keys

84 Evo: blue 2nd key, green alpha key, white math, arithmetic, and function keys, black arrow and number keys

number keys: [ 0 ] through [ 9 ], [ . ], [(-)]

Connection Software:

84 Python: TI Connect CE

84 Evo: https://connectevo.ti.com/ticevo/en/ (online connection, like TI-nSpire CX II)

Multiplication Symbol Used in Expressions:

84 Python: asterisk (*)

84 Evo: dot (⋅)

Colors Used on Calculations on the Home/Calculator Screen:

84 Python: Everything is in black

84 Evo: Input expressions are in black, cursor is blue, answers are in green

Memory Available for Storage:

84 Python: 3 MB

84 Evo: 3.5 MB

Graphing Display Area:

84 Python: 264 x 165 pixels with a border around the graphing

84 Evo: 319 x 209 with no boarder as the graph takes the entire screen (just like the old days, the monochrome TI-84)

Backwards Capability of TI-83 Plus and TI-84 Plus TI-Basic Programs:

84 Python: Yes (.8xp)

84 Evo: No (.8xp2). Why? Texas Instruments rewrote the Basic language engine. There are programs that can translate from .8xp to .8xp2 files. Check Cemetech or TI Planet.

Note: Python programs can be transferred easily between the 84 Python and 84 Evo.

Built in Clock:

Python: Set clock option present

Evo: No clock option present (quiet removal, will the clock be missed?)

Features Added to the TI-84 Evo

* Gradian mode, with all the conversions and symbols added

* Three additional regressions: PropReg (y = a*x), RecipReg (y = a/x + b), and eBaseReg regression (y = a * e^(b*x))

* The Time Module adds two additional functions: ticks_ms and ticks_diff.

Should I Get a TI-84 Evo or TI-84 CE Python?

Reasons to buy the TI-84 Evo:

You like connecting with USB C instead of USB Mini

Faster processor*

Your first TI-84 ever

You desire to work with both hardware and the emulator (you get a four year license on purchase)

You want the latest model and/or if you are like me and like to collect calculators

Reasons to buy the now older TI-84 CE Python:

The border around the graphing screen isn't annoying, don’t mind it as much (or at all)

You work with the SciTools and Periodic Table apps a lot

You want a TI-84 with Python at a lower price

You work with the Turtle Module in Python

* although some say graphing could be slower due to the fact the entire screen is used.

Note: If you want assembly programming with a TI, stick with the older monochrome calculators as the newer TI calculators no longer support assembly programs.

Source

Texas Instruments. “TI-84 Evo Graphing Calculator” https://education.ti.com/en/products/calculators/graphing-calculators/ti-84-evo Retrieved May 23, 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.

HP 12C Platinum: Present Value of a Fractional Year

 HP 12C Platinum: Present Value of a Fractional Year

This blog features the HP 12C Platinum, HP 10BII+, and HP 22S calculators.

Short Term Transactions

Here is the scenario: A bank offers a short term bond, which last less than one year, which pays $100.00 at maturity date. The interest rate stated is an annual interest rate. While determining a pricing schedule, one banker uses an HP 12C Platinum calculator while another uses the HP 10BII+ calculator. They both use the TVM (time value of money) keys. A 365-day year is used.

FV = -100, I% (see table), N (see table), PMT = 0, Solve for PV, P/Y = 1

Term (days)	N = term ÷ 365 (to five decimal places) (for reference)	I%	HP 12C Platinum (to 5 decimal places)	HP 10BII+ (to 5 decimal places)
89	0.24384	5	98.79551	98.81737
141	0.38630	5	97.99973	98.02800
181	0.49589	5	97.58054	97.60958
365	1	5	95.23810	95.23810
89	0.24384	8	98.08664	98.14091
141	0.38630	8	96.83753	96.90714
181	0.49589	8	96.18425	96.25548
365	1	8	92.59259	92.59259

As you can see, the results are different! Why?

According to HP-12C Solutions Handbook (see the Source section), when it comes to fractional periods, simple interest is used instead of compound interest in the TVM solver. Most financial calculators, such as HP 10BII+ always uses compound interest.

Cash flow convention states that:

1. Cash inflows, such as deposits, are positive.

2. Cash outflows, such as payments, are negative.

3. In most problems, the present value and future value have opposite signs.

Respecting cash flow convention, the formulas for present value are:

Simple Interest:

P = -F ÷ (1 + D ÷ 365 × I ÷ 100)

Compound Interest:

P = -F ÷ (1 + I ÷ 100) ^ (D ÷ 365)

where:

P = present value (PV)

F = future value (FV)

I = annual interest rate

D = number of days

If leap years, substitute 366 for 365. If we are working with 30/360 day years, substitute 360 for 365.

These formulas are set up to be entered in calculators with equation solvers such as the HP 22S. I have used the HP 22S to verify each of the results above.

Now why is the results the say when the term exactly 365? It’s pretty simple to prove:

Simple Interest:

P_simple = -F ÷ (1 + 365 ÷ 365 × I ÷ 100) = -F ÷ (1 + I ÷ 100)

Compound Interest:

P_compound = -F ÷ (1 + I ÷ 100) ^ (365 ÷ 365) = -F ÷ (1 + I ÷ 100) = P_simple

When the Term Exceeds One Year

Let’s say the $100.00 bond lasts for 545 days, about one year and a half. This time the interest rate is 7%.

On the HP 12C, any fractional period is treated with simple interest. The HP 12C’s TVM solver (and the HP 12C Platinum) treats the timeline as such.

365 days: full period, compound interest	180 days: partial year, simple interest
PV	FV = -$100.00

To break it down, the HP 12C starts determining the value after 365 days.

N = 180 ÷ 365

I = 7

FV = -100

PV ≈ 96.66314

P = -(-100) ÷ (1 + 180 ÷ 365 × 7 ÷ 100) ≈ 96.66314

365 days: full period, compound interest	180 days: partial year, simple interest
PV	FV = -$100.00
	96.66314

From here, the HP 12C uses that value to calculate final present value. Since we are now working with a full period (one year in this case), compound interest is used with n = 1:

N = 1

I = 7

FV ≈ -96.66314 (treated as an outflow and becoming the acting future value)

PV ≈ -(-96.6314 ÷ (1 + 7 ÷ 100) ^ (1) ≈ 90.33938

The final present value (and price) of this bond is 90.33938.

If we enter following the HP 12C Platinum:

N: 545 [ ENTER ] 365 [ ÷ ] [ N ] (≈ 1.49315)

I: 7 [ i ]

FV: 100 [ CHS ] [ FV ]

PMT: 0 [ PMT ]

[ PV ] → PV ≈ 90.33938

Enter the same problem on most other financial calculators, like the HP 10BII+, will result in a final present value of 90.39108. (P/Y = 1) This is because compounding interest is used for the entire time:

P = -(-100) ÷ (1 + 7 ÷ 100) ^ (545 ÷ 365) ≈ 90.39108

HP 12C Program: Present Value Using Compounding Interest Including Fractional Periods

The program calculates present value given the future value, interest, and the number of days using compounding interest for the entire period. A 365 day year is assumed.

Code: Key; Key Code

ENTER; 36

3; 3

6; 6

5; 5

÷; 10

1; 1

RCL i; 45, 12

%; 25

+; 40

x<>y; 34

y^x; 21

RCL FV; 45, 15

x<>y; 34

÷; 10

CHS; 16

GTO 000; 43,33,000 (GTO 00; 43, 33,00 for HP 12C Classic)

Future value is stored in FV and interest rate is stored in i. The number of days is on the X stack.

Source

Hewlett Packard. HP-12C Solutions Handbook. 2004. pg. 45 https://literature.hpcalc.org/official/hp12c-sh-en.pdf

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.