RPN: HP 32S/DM32: Simple Harmonic Motion
Introduction
The simple harmonic motion (SHM) of a spring can be described by Hooke’s Law:
F = -K * x(t)
x(t) = position of the spring
F = force of the spring
Note that: F = mass * acceleration = M * x’’(t)
Then:
M * x’’(t) = - K * x(t)
M * x’’(t) + K * x(t) = 0
x’’(t) + (K / M) * x(t) = 0
Without going into much details, two general solutions can have the form:
Case I:
x(t) = A * sin(W * t + B) with W = √(K / M) (A, B are constants)
and
Case II:
x(t) = A * cos(W*T + D) with W = √(K / M) (A, D are constants)
Believe it or not, both solutions work.
Case I:
x(t) = A * sin(W * t + B)
x’(t) = A * W * cos(W * t + B)
x’’(t) = -A * W^2 * sin(W * t + B)
Observe:
x’’(t) = -A * W^2 * sin(W * t + B)
x’’(t) = -W^2 * (A * sin(W * t + B))
x’’(t) = -W^2 * x(t)
W = √(K / M), W^2 = K / M
x’’(t) = -(K / M) * x(t)
M * x’’(t) = -K * x(t)
We are back to Hooke’s Law.
Case II:
x(t) = A * cos(W * t + D)
x’(t) = -A * W * sin(W * t + D)
x’’(t) = -A * W^2 * cos(W * t + D)
Then:
x’’(t) = -W^2 * (A * cos(W * t + D))
x’’(t) = -W^2 * x(t)
x’’(t) = -(K / M) * x(t)
M * x’’(t) = -K * x(t)
We once again arrive at Hooke’s Law. Another valid solution.
These two solutions are chosen for simplicity. A general solution would take the form of:
x(t) = A2 * cos(W * t + D) + A2 * sin(W * t + B)
Determining the Constants
Let:
x0 = initial position, at t = 0
v0 = initial velocity, at t = 0
Let’s use the simplified solutions.
Case I: Sine Approach
x(t) = A * sin(W * t + B)
x’(t) = A * W * cos(W * t + B)
x0 = A * sin(B)
v0 = A * W * cos(B)
x0 = A * sin(B)
v0 / W = A * cos(B)
Square each equation and add:
x0^2 + (v0 / W)^2 = A^2 * ((sin B)^2 + (cos B)^2)
x0^2 + (v0 / W)^2 = A^2 * 1
√( x0^2 + (v0 / W)^2 ) = A
A = √( x0^2 + (v0 / W)^2 )
Divide the top equation by the bottom equation:
x0 / (v0 / W) = tan(B) (assume A ≠ 0)
B = arctan( x0 / (v0 / W) )
⇒ atan2( x0, (v0 / W)) = arg( (v0 / W) + x0*i ) where i = √-1
Case II: Cosine Approach
x(t) = A * cos(W * t + D)
x’(t) = -A * W * sin(W * t + D)
x0 = A * cos(D)
v0 = -A * W * sin(D)
x0 = A * cos(D)
v0 / W = -A * sin(D)
Square each equation and add:
x0^2 + (v0 / W)^2 = (A * cos(D))^2 + (-A * sin(D))^2
x0^2 + (v0 / W)^2 = A^2 * cos(D)^2 + (-A) *(-A) * sin(D)^2
x0^2 + (v0 / W)^2 = A^2 * cos(D)^2 + A^2 * sin(D)^2
x0^2 + (v0 / W)^2 = A^2 * 1
√( x0^2 + (v0 / W)^2 ) = A
A = √( x0^2 + (v0 / W)^2 )
Divide the top equation by the bottom equation:
x0 / (v0 / W) = -cot(D) (assume A ≠ 0)
(v0 / W) / x0 = -tan(D)
-(v0 / W) / x0 = tan(D)
D = arctan(-(v0 / W) / x0)
⇒ atan2( -(v0 / W), x0) = arg( x0 - (v0 / W)*i)
The atan2, arg, and similarly, the rectangular to polar conversion functions are used to allow for all angles, including angles in the form of π/2 + n*π.
The following code demonstrates the two basic approaches, using the classic HP 32 (and which can be used on HP 32SII, DM32, HP 33S).
HP 32S Code
The variables:
X = (initial) position of the spring. When the spring is at it’s natural or neutral position, X = 0. (SI: unit m)
V = (initial) velocity of the spring. If V<0, the spring is getting shorter. If V>0, the spring is getting longer.
M = mass of the object attached to the end of the string (SI: unit kg)
K = spring’s force constant (N/m, kg/s^2)
A = amplitude of the spring (m)
Labels:
LBL I: Initialization, calculate W = √(K / M), and A = √(X^2 + V^2/W^2)
(HP 32S Memory: I: 34.5 bytes)
I01 LBL I
I02 INPUT K
I03 INPUT M
I04 INPUT X
I05 INPUT V
I06 RAD
I07 RCL K
I08 RCL÷ M
I09 SQRT
I10 STO W
I11 VIEW W
I12 RCL X
I13 x^2
I14 RCL V
I15 x^2
I16 RCL W
I17 x^2
I18 ÷
I19 +
I20 SQRT
I21 STO A
I22 VIEW A
I23 RTN
Sine Approach: x(t) = A * sin(W*T + B); W = √(K / M)
LBL S: Calculate B, then calculates X from time = 0 seconds to T seconds (per second)
(HP 32S Memory: S: 21.0 bytes, U: 16.5 bytes)
S01 LBL S
S02 RCL X
S03 RCL V
S04 RCL W
S05 ÷
S06 y,x→Θ,r
S07 x<>y
S08 STO B
S09 VIEW B
S10 INPUT T
S11 3
S12 10^x
S13 ÷
S14 STO T
U01 LBL U
U02 RCL T
U03 IP
U04 RCL× W
U05 RCL+ B
U06 SIN
U07 RCL× A
U08 STOP
U09 ISG T
U10 GTO U
U11 RTN
Cosine Approach: x(t) = A * cos(W*T + D); W = √(K / M)
LBL C: Calculate D
LBL D: Calculate X from time = 0 seconds to T seconds (per second)
(HP 32S Memory: C: 22.5 bytes, U: 16.5 bytes)
C01 LBL C
C02 RCL V
C03 RCL W
C04 ÷
C05 +/-
C06 RCL X
C07 y,x→Θ,r
C08 x<>y
C09 STO D
C10 VIEW D
C11 INPUT T
C12 3
C13 10^x
C14 ÷
C15 STO T
E01 LBL E
E02 RCL T
E03 IP
E04 RCL× W
E05 RCL+ D
E06 COS
E07 RCL× A
E08 STOP
E09 ISG T
E10 GTO E
E11 RTN
Examples
The calculator is set to FIX 5 mode.
Example 1:
K = 40, M = 1.33, V = 0, X = 0.3. Up to 5 seconds
Results:
W = 5.48408, A = 0.30000
|
Sine Approach (LBL S) |
Cosine Approach (LBL C) |
Time (seconds) |
B = 1.57080 (π/2) |
D = 0 |
0 |
0.30000 |
0.30000 |
1 |
0.20921 |
0.20921 |
2 |
-0.00822 |
-0.00822 |
3 |
-0.22067 |
-0.22067 |
4 |
-0.29955 |
-0.29955 |
5 |
-0.19711 |
-0.19711 |
Example 2:
K = 48, M = 3, V = 0, X = -0.5. Up to 5 seconds
Results:
W = 4.00000, A = 0.50000
|
Sine Approach (LBL S) |
Cosine Approach (LBL C) |
Time (seconds) |
B = -1.57080 (-π/2) |
D = 3.14159 (π) |
0 |
-0.50000 |
-0.50000 |
1 |
0.32682 |
0.32682 |
2 |
0.07275 |
0.07275 |
3 |
-0.42193 |
-0.42193 |
4 |
0.47883 |
0.47883 |
5 |
-0.20404 |
-0.20404 |
Sources
“Simple Harmonic Motion” Hyperphysics. http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html Retrieved February 7, 2025.
LibreTexts Physics. “15.2: Simple Harmonic Motion” https://phys.libretexts.org/Bookshelves/University_Physics/University_Physics_(OpenStax)/Book%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/15%3A_Oscillations/15.02%3A_Simple_Harmonic_Motion Retrieved February 6, 2025.
Rodicek, Danny. Mathematics & Physics for Programmers Hingham, MA. 2005. ISBN 1-58450-330-0
Eddie
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