Problem 48 - Calculus III

Question

Suppose that the position function for an object in three dimensions is given by the equation \(\mathbf{r}(t)=t \cos (t) \mathbf{i}+t \sin (t) \mathbf{j}+3 t \mathbf{k}\) Show that the particle moves on a circular cone.

Step-by-Step Solution

Answer

The particle moves on a cone as \( z = 3\sqrt{x^2 + y^2} = 3t \), satisfying the cone equation.

Step 1: Analyze the Position Vector

The position function is given by \( \mathbf{r}(t) = t \cos(t) \mathbf{i} + t \sin(t) \mathbf{j} + 3t \mathbf{k} \). This represents a curve parameterized by \( t \) in 3D space where each component corresponds to \( x(t) = t \cos(t) \), \( y(t) = t \sin(t) \), and \( z(t) = 3t \).

Step 2: Identify the Components for the Cone Equation

For a particle to move on a cone, its position must satisfy a cone equation. A common form for the equation of a circular cone aligned along the z-axis is \( z = c \sqrt{x^2 + y^2} \) where \( c \) is a constant.

Step 3: Express x and y as a Function of t

We have \( x(t) = t \cos(t) \) and \( y(t) = t \sin(t) \). Therefore, \( x^2 + y^2 = (t \cos(t))^2 + (t \sin(t))^2 = t^2 (\cos^2(t) + \sin^2(t)) = t^2 \).

Step 4: Relate z to x and y

We know that \( z(t) = 3t \). From Step 3, we found that \( x^2 + y^2 = t^2 \). Now we check if \( z(t) \) and \( \sqrt{x^2 + y^2} \) satisfy the cone equation: \( z = 3\sqrt{x^2 + y^2} = 3t \).

Step 5: Verify the Cone Equation

Since \( z(t) = 3t \) and \( 3\sqrt{x^2 + y^2} = 3t \), we see that these are equal, confirming that the particle moves on a circular cone.

Key Concepts

Parametric Equations in Calculus

In calculus, parametric equations offer a robust way to describe curves that traditional Cartesian equations might struggle to capture, especially in multidimensional settings. Unlike regular equations where a single equation relates two variables, parametric equations use one or more parameters to define a set of related quantities. In three dimensions, these equations become especially useful.

Imagine wanting to describe the path of an object flying through space. Using a parameter, often the variable is time, allows us to track the object’s journey by focusing on how each coordinate changes with time. For example, based on the equation \( \mathbf{r}(t) = t \cos(t) \mathbf{i} + t \sin(t) \mathbf{j} + 3t \mathbf{k} \), we see:

• \( x(t) = t \cos(t) \) describes the horizontal movement.
• \( y(t) = t \sin(t) \) accounts for the vertical component.
• \( z(t) = 3t \) represents the elevation.

Each of these parts is a function of the parameter \( t \), giving a clear picture of how our object moves through three-dimensional space as \( t \) changes.

Conic Sections

Conic sections are a central theme in math involving curves formed by intersecting a cone with a plane in various ways.

These shapes include circles, ellipses, parabolas, and hyperbolas. While it may initially seem that conic sections are restricted to flat surfaces, their equations hold great power in understanding spatial curves.

In 3D space, these sections take on new life. For our current exploration, we're interested in the circular cone, which is created by rotating a line around an axis to form a cone-like shape. Imagine this setup:

• A set of lines passing through a single point (the vertex), opening up energy as they rotate.
• When a plane intersects this shape horizontally, we get a circle.
• Tilting the plane results in an ellipse or other conic forms.

The particle's movement, as described by our position function, forms a circular path revolving around a central axis - the essence of circular motion on a cone.

Position Vector Analysis

Position vectors are like handy arrows that help us pinpoint an object's location in space. Instead of struggling with which axis a point lies on, vectors describe a position using direction and magnitude.

In the context of our example, the position vector \( \mathbf{r}(t) = t \cos(t) \mathbf{i} + t \sin(t) \mathbf{j} + 3t \mathbf{k} \) is crucial for understanding motion through space. Each component tells a little story about how the object travels:

• The \( x \) and \( y \) components, \( t \cos(t) \) and \( t \sin(t) \), capture lateral movement. Observe how they resemble familiar trigonometric functions, connecting insights back to circular motion.
• The \( z \) component \( 3t \) suggests motion steadily upward, adding depth to motion analysis.

By working through these steps, we see that the object's position at any time \( t \) provides insight into how it traces out a curve, passing through points on a circular cone as it progresses.