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Harmonic Motion with Damping

Harmonic Motion with Damping
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2020-02-28 22:08
FunctionTrigonometryCalculus
This example demonstrates damped harmonic motion. It involves differential equations and trigonometric functions from calculus. By constructing objects like function f, slider animation button1, and loci C_c, C_u, the dynamic process of harmonic motion is clearly illustrated. Suitable for high school physics and mathematics courses.

GeoGebra Commands & Parameters

  1. Numeric y_0(1)
  2. Numeric v_0(5)
  3. Numeric p(0.5)
  4. Numeric q(16)
  5. ((-p) - sqrt(p^(2) - (4 * q))) / 2
  6. ((-p) + sqrt(p^(2) - (4 * q))) / 2
  7. ((-v_0) + (l_1 * y_0)) / (l_1 - l_2)
  8. (v_0 - (l_2 * y_0)) / (l_1 - l_2)
  9. ((-p)) / 2
  10. sqrt((4 * q) - p^(2)) / 2
  11. y_v(x) = (C_1 * ℯ^((l_1 * x))) + (C_2 * ℯ^((l_2 * x)))
  12. Boolean d(true)
  13. f(x) = x^(2) + (p * x) + 16
  14. (y_0 * p / 2) + v_0
  15. (v_0 - (y_0 * a)) / b
  16. y_u(x) = (ℯ^((a * x)) * ((y_0 * cos((b * x))) + (C_u * sin((b * x)))))
  17. y_c(x) = (ℯ^((l_1 * x)) * (y_0 + (C_c * x)))
  18. 5

AI Drawing Prompt Reference

Draw a damped harmonic motion graph, including a quadratic function f(x) = x^2 + 0.5x + 16 with initial conditions y_0=1 and v_0=5. Adjust parameters p and q to observe changes in functions y_v(x) and y_u(x), where y_v(x) = (C_1 * e^(l_1 * x)) + (C_2 * e^(l_2 * x)) and y_u(x) = (e^(a * x) * (y_0 * cos(b * x) + C_u * sin(b * x))). Also, plot y_c(x) = (e^(l_1 * x) * (y_0 + C_c * x)) and observe its changes over time.
The prompt above was compiled from the original drawing commands and is for reference only.