**HP 48G: Scientific Applications**

Calculators Featured: HP 48G, HP 48G+, HP 48GX

**Theoretic Freezing Level**

The program FRZLVL calculates the theoretical freezing level in feet above sea level given the outside temperature at a given altitude. The results are displayed in feet (the feet unit is attached).

**HP 48G Program: FRZLVL**

<< "FREEZING LEVEL"

{ { "ALT:" "DEFAULT FT" }

{ "TEMP:" "DEFAULT °C" } }

{ } { } { }

IF INFORM

THEN OBJ→ DROP DUP TYPE

IF 13 == THEN '1_°C' CONVERT

ELSE '1_°C' →UNIT

END SWAP DUP TYPE

IF 13 == THEN '1_ft' CONVERT

ELSE '1_ft' →UNIT

END SWAP '1000_ft/°C'

* DUP2 2 / +

"DRY" →TAG 3 ROLLD

1.5 / + "WET" →TAG END >>

**Instructions**

1. Run FRZLVL

2. ALT: Enter altitude. You can attach units if you want, the default is altitude is in feet.

3. TEMP: Enter outside air temperature. You can attach units if you want, the default unit is degrees Celsius (°C).

4. Results: freezing dry level in feet and freezing wet level in feet.

**Example**

Altitude: 1,750 ft

Temperature: -2°C (28.4°F)

Results:

DRY: 750_ft

WET: 416.666666667_ft

Source:

Hewlett Packard "Predicting Freezing Levels" HP 65 Aviation Pac 1. Hewlett Packard. 1974

**Lens Calculations **

**Variables**

Please note the inequality restrictions.

R = radius of curvature

D = diameter of the lens, D ≤ 2*R

S = sag S ≤ R

θ = tangent angle between lens axis and reflection

**HP 48G Program: LENSANG**

<< 'R=(D^2+4*S^2)/(8*S)' STEQ

{ { "R/D" << 'D' STO 'R' STO EQ 'S' 0 ROOT 'S' →TAG >> }

{ "R/S" << 'S' STO 'R' STO EQ 'D' 0 ROOT 'D' →TAG >> }

{ "D/S" << 'S' STO 'D' STO EQ 'R' 0 ROOT 'R' →TAG >> }

{ "→θ" << 'D' RCL 'R' RCL 'S' RCL - 2 * / ATAN 'θ' DUP2 STO →TAG >> } }

TMENU >>

**Instructions**

1. Run LENSANG. A temporary custom menu appears.

2. Enter two known amounts and press the appropriate key to store the variables:

If R and D are known: R [ ENTER ] D [ F1 ]

If R and S are known: R [ ENTER ] S [ F2 ]

If D and S are known: D [ ENTER ] S [ F3 ]

The third variable between R, D, and S is calculated and displayed as a result.

3. Press [ F4 ] to calculate the angle, θ.

Results are stored in the variables R, S, D, and θ.

**Examples**

Known: R = 5.8, D = 7.6

LENSANG 5.8 [ ENTER ] 7.6 [ F1 ] (R/D)

Result: S: 1.41821953996

[ F4 ] (θ). Result: θ: 40.9327245742

Known: D = 11, S = 9

LENSANG 11 [ ENTER ] 9 [ F3 ] (D/S)

Result: R: 6.18055555556

[ F4 ] (θ). Result: θ: -62.8591312297

Source:

Tuchscherer, L.D. "Lens Calculations - SAG, ANGLE, MIN/MAX" HP 67 Optics Hewlett Packard. 1978.

**Period of a Pendulum**

The program PENDULUM calculates the period of a pendulum given two parameters:

ANGLE: the angle that the pendulum swings out. The angle must be entered in radians.

LENGTH: The length of the pendulum in meters.

Unlike most calculations, the program uses an Legendre elliptic integral of the first kind to increase accuracy.

T = 4 * √(L / g) * ∫( dx / √(1 - k^2 * sin^ x), x, 0, π/2) where k = sin(α/2)

g = Earth's gravitational constant = 9.80665 m/s^2

This program does not use units.

**HP 48G Program: PENDULUM**

<< RAD

"PENDULUM"

{ { "ANGLE:" "IN RADIANS" }

{ "LENGTH:" "IN METRES" } }

{ } { } { }

IF INFORM

THEN OBJ→ DROP

9.80665 / √ 4 *

SWAP 2 / SIN SQ 0 θ π 2 /

→NUM 3 ROLL 'X'

SIN SQ * 1 SWAP - √

INV 'X' ∫ →NUM *

"PERIOD" →TAG END >>

**Example**

ANGLE: π/6 (enter this as 'π/6')

LENGTH: 1.5 (m)

Result:

PERIOD: 2.50011881685 (s)

Source:

Steers, Hugh. "The Pit & Pendulum" Datafile. ISSN 1352-8254. Handheld and Portable Computer Club. V29 N1. January - March 2010

Eddie

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