Ch.12 - Parametric and Polar Curves

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch.12 - Parametric and Polar Curves

Problem 12.1.54

Chapter 12, Problem 12.1.54

53–56. Circular motion Find parametric equations that describe the circular path of the following objects. For Exercises 53–55, assume (x, y) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There are many ways to describe any circle.

The tip of the 15-inch second hand of a clock completes one revolution in 60 seconds.

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Identify the radius of the circular path, which is the length of the second hand. Here, the radius \(r\) is 15 inches.

Determine the angular velocity \(\omega\) of the second hand. Since it completes one full revolution (which is \(2\pi\) radians) in 60 seconds, calculate \(\omega\) as \(\omega = \frac{2\pi}{60}\) radians per second.

Set up the parametric equations for the circular motion. The general form for a circle centered at the origin is \(x(t) = r \cos(\omega t)\) and \(y(t) = r \sin(\omega t)\), where \(t\) is time in seconds.

Substitute the known values of \(r\) and \(\omega\) into the parametric equations to express \(x(t)\) and \(y(t)\) explicitly in terms of \(t\).

Interpret the parametric equations: as \(t\) increases from 0 to 60, the point \((x(t), y(t))\) traces the circular path of the second hand exactly once.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Parametric Equations for Circular Motion

Parametric equations express the coordinates of a point on a circle as functions of time, typically using sine and cosine functions. For a circle centered at the origin with radius r, the position (x, y) can be described as x = r cos(θ(t)) and y = r sin(θ(t)), where θ(t) is the angle as a function of time.

Angular Velocity and Period

Angular velocity (ω) measures how fast an object rotates, defined as the angle covered per unit time. It relates to the period (T), the time for one full revolution, by ω = 2π / T. For the clock's second hand, completing one revolution in 60 seconds means ω = 2π / 60 radians per second.

Relationship Between Linear and Angular Quantities

The linear position on the circle depends on the radius and the angular position. The radius is the length from the center to the point (here, 15 inches), and the angular position changes over time according to angular velocity. This relationship allows converting rotational motion into x and y coordinates.

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Related Practice

Textbook Question

84. Arc length for polar curves: Prove that the length of the curve r = f(θ) for α ≤ θ ≤ β is

L = ∫(α to β) √(f(θ)² + f'(θ)²) dθ.

Textbook Question

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.

x = 2 + √t, y = 2 - 4t; slope = -8

Textbook Question

57–64. Graphing polar curves Graph the following equations. Use a graphing utility to check your work and produce a final graph.

r = sin 3θ

Textbook Question

15–30. Working with parametric equations Consider the following parametric equations.

a. Eliminate the parameter to obtain an equation in x and y.

b. Describe the curve and indicate the positive orientation.

x = √t + 4, y = 3√t; 0 ≤ t ≤ 16

Textbook Question

45–60. Areas of regions Find the area of the following regions.

The region inside the curve r = √(cos θ) and inside the circle r = 1/√2 in the first quadrant

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Textbook Question

13–30. Graphing conic sections Determine whether the following equations describe a parabola, an ellipse, or a hyperbola, and then sketch a graph of the curve. For each parabola, specify the location of the focus and the equation of the directrix; for each ellipse, label the coordinates of the vertices and foci, and find the lengths of the major and minor axes; for each hyperbola, label the coordinates of the vertices and foci, and find the equations of the asymptotes.

10x² - 7y² = 140

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