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Modeling with Parametrically Defined Curves

Explore examples of parametric equations for circular movements including clock hands, planetary orbits, and pendulum swings. Learn how to model and calculate these dynamic systems in two dimensions.

kennedy-little
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Modeling with Parametrically Defined Curves

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  1. Modeling with Parametrically Defined Curves Section 10.3

  2. Example 1 • The Colgate Clock in Jersey City is the world’s largest free-standing clock. It is a circle 50 feet in diameter and the bottom of its face is about 14 feet above the ground. Consider a coordinate system on the ground under the center of the clock. Write the parametric equations fo rhte location of a minute hand after t minutes when t =0 is midnight.

  3. Example 1 • Key info: • Clock – so clockwise! • Starts at the top of the graph • Diameter is 50 feet • Vertical displacement: 14 feet So x(t) = sint y(t) = cos t Amp 50/2 = 25 Start: 39 feet Period : 2π/60

  4. Example 2 • Suppose a planet is orbiting a star in a circular orbit 200 million miles from the center of the star. Suppose a moon is circling the planet 30 times during the orbit of the planet. Give possible parametric equations for the moon.

  5. Example 2 • First do the planet to the star • Assume counterclockwise x(t) = 200cos(t) Y(t) = 200 sin(t) Now, moon to planet x(t) = 1.5cos(30t) y(t) = 1.5 sin(30t) x(t) = 200 cos(t) + 1.5cos(30t) y(t) = 200sin(t) + 1.5sin(30t)

  6. Example 3 • Find a set of parametric equations to model the movement in two dimensions of an 18-inch long pendulum, which swings through a 45˚ arc in about 1.4 seconds. Draw a picture! Period: 2π/2.8 Split into 2 angles: 22.5 18*Sin 22.5 = 6.8883  amp

  7. Homework Pages 597 – 598 1 – 6, 8

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