Problem 54 - Calculus Early Transcendentals: Pearson New International Edition

Question

The ends of an elastic string with a knot at \(K(x, y)\) are attached to a fixed point \(A(a, b)\) and a point \(P\) on the rim of a wheel of radius \(r\) centered at \((0,0)\). As the wheel turns, \(K\) traces a curve \(C\). Find the equation for \(C\). Assume that the string stays taut and stretches uniformly (i.e., \(\alpha=|K P| /|A P|\) is constant).

Step-by-Step Solution

Answer

The curve \(C\) is described by \((x - r\cos\theta)^2 + (y - r\sin\theta)^2 = \alpha^2 ((x - a)^2 + (y - b)^2)\), for a constant \(\alpha\).

Step 1: Express Knot Distance to Wheel Center

Given the point on the rim, due to the circle's characteristics, we can express the position of point \(P\) as \( (r\cos\theta, r\sin\theta) \), where \(\theta\) is the angular position of the point on the wheel. Thus, the distance from \(K(x,y)\) to \(P\) can be expressed as \( \sqrt{(x - r\cos\theta)^2 + (y - r\sin\theta)^2} \).

Step 2: Express Knot Distance to Fixed Point

The distance from the knot \(K\) to the fixed point \(A(a, b)\) can be expressed as \( \sqrt{(x - a)^2 + (y - b)^2} \).

Step 3: Relate the Distances with Constant Alpha

Given that the string stretches uniformly, we know that \( \alpha = \frac{|KP|}{|AP|} \), meaning \( \alpha = \frac{\sqrt{(x - r\cos\theta)^2 + (y - r\sin\theta)^2}} {\sqrt{(x - a)^2 + (y - b)^2}} \).

Step 4: Solve for the Curve Equation

Squaring both sides to eliminate the square roots, we have:\[(x - r\cos\theta)^2 + (y - r\sin\theta)^2 = \alpha^2 ((x - a)^2 + (y - b)^2)\]This is the implicit equation that \(K(x,y)\) follows as the point \(P(r\cos\theta, r\sin\theta)\) traces the circle. Simplifying further will give the specific relationship of the parameters based on the point rotation \( \theta \).

Step 5: Consider Simplification or Specific Cases

In some specific cases or simplifications depending on known values of \(a\), \(b\), or \(\alpha\), additional simplifications to the equation may be possible. However, generally, the form reached in Step 4 is sufficient to describe the locus of \(K\).

Key Concepts

Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and properties of space. It plays a crucial role in solving problems related to figures like circles, triangles, and polygons. In this particular problem, understanding the geometry of circles and the position of points on a plane helps set up the equations to describe motion.
- A circle's key characteristic is that any point on its rim is equidistant from the center. - The position of point on the circle can be expressed in terms of angles, using trigonometric functions such as cosine and sine.
Knowing these basics aids in expressing the position of any moving point on the circle, crucial for solving problems like finding the locus of moving points.

Locus of a Point

The locus of a point in mathematics refers to all possible positions that a point can have given certain conditions. It is essentially the path traced by a moving point under specified constraints.
- In this problem, the locus of the point is the path traced by the knot K as the wheel rotates. - To find the locus, we explore how the coordinates of K change with the angle in circular motion. - The locus can be described by an equation that incorporates the conditions under which the point moves, including the distances described by the problem statement.
Understanding the locus allows us to predict and describe the movement pattern mathematically.

Parametric Equations

Parametric equations express the coordinates of the points that form a curve or a surface in terms of a single parameter, usually denoted as \( \theta \) for angles.
- These equations are particularly useful for representing curves and motion, as they simplify expressing the position of a point as it varies with the angle or time.- In the circular motion problem, we use parametric equations to describe the position of point \( P \) on the circle: \( (r\cos\theta, r\sin\theta) \).- By relating these parameters with conditions like symmetry and distance, we can develop a more precise equation for the locus of point K.
This method provides a structured way to solve complex motion problems by breaking them down into simpler, manageable equations tied to a single changing variable.

Circular Motion

Circular motion refers to movement along the circumference of a circle. It is a common topic in physics and mathematics, used to understand rotational dynamics.
- In this exercise, point \( P \) moves along the rim of a wheel, which is a classic example of circular motion.- The movement is characterized by changing angles, usually denoted as \( \theta \), and can be described using parametric equations.- As the wheel turns, the knot K traces a unique path influenced by both the rotation and the stretching of the elastic string.
Understanding circular motion requires knowing the relationships between angle, radius, and the resulting position on the circle, which are pivotal in calculating the precise locus of K.