Problem 47 - Calculus Early Transcendentals: Pearson New International Edition

Question

Describe in general terms the following "helical" type motions: (a) \(\mathbf{r}(t)=\sin t \mathbf{i}+\cos t \mathbf{j}+t \mathbf{k}\) (b) \(\mathbf{r}(t)=\sin t^{3} \mathbf{i}+\cos t^{3} \mathbf{j}+t^{3} \mathbf{k}\) (c) \(\mathbf{r}(t)=\sin \left(t^{3}+\pi\right) \mathbf{i}+t^{3} \mathbf{j}+\cos \left(t^{3}+\pi\right) \mathbf{k}\) (d) \(\mathbf{r}(t)=t \sin t \mathbf{i}+t \cos t \mathbf{j}+t \mathbf{k}\) (e) \(\mathbf{r}(t)=t^{-2} \sin t \mathbf{i}+t^{-2} \cos t \mathbf{j}+t \mathbf{k}, t>0\) (f) \(\mathbf{r}(t)=t^{2} \sin (\ln t) \mathbf{i}+\ln t \mathbf{j}+t^{2} \cos (\ln t) \mathbf{k}, t>1\)

Step-by-Step Solution

Answer

Each vector describes a type of helical motion with varying spiraling behavior and axis orientations.

Step 1: Analyze component functions in (a)

The vector function for (a) is \( \mathbf{r}(t)=\sin t \mathbf{i}+\cos t \mathbf{j}+t \mathbf{k} \). The components \( \sin t \) and \( \cos t \) show a circular motion in the \(xy\)-plane since they describe a circle of radius 1. The \(t\) component in the \(z\)-axis represents a linear progression upwards as \(t\) increases. Therefore, this describes a helix oriented around the \(z\)-axis.

Step 2: Analyze component functions in (b)

For (b), the vector is \( \mathbf{r}(t)=\sin t^{3} \mathbf{i}+\cos t^{3} \mathbf{j}+t^{3} \mathbf{k} \). Similar to (a), \( \sin t^{3} \) and \( \cos t^{3} \) suggest circular motion in the \(xy\)-plane, but the angular speed increases rapidly because of the \(t^{3}\) argument. The \(t^{3}\) term in the \(z\)-axis implies an accelerating upward motion, still forming a helix but with stretch.

Step 3: Analyze component functions in (c)

In (c), the vector function is \( \mathbf{r}(t)=\sin (t^{3} + \pi) \mathbf{i} + t^{3} \mathbf{j} + \cos (t^{3} + \pi) \mathbf{k} \). The expressions \( \sin (t^{3} + \pi) \) and \( \cos (t^{3} + \pi) \) create a circular path in the \(xk\)-plane. The \(y\)-coordinate \(t^{3}\) grows rapidly, resulting in an increasing vertical displacement. This motion is helical around the \(y\)-axis.

Step 4: Analyze component functions in (d)

In (d), \( \mathbf{r}(t)=t \sin t \mathbf{i} + t \cos t \mathbf{j} + t \mathbf{k} \), the terms \(t \sin t\) and \(t \cos t\) suggest a spiral-out pattern in the \(xy\)-plane. The height \(t\) increases linearly, forming a conical helix that spirals outward and upward.

Step 5: Analyze component functions in (e)

For (e), we have \( \mathbf{r}(t) = t^{-2} \sin t \mathbf{i} + t^{-2} \cos t \mathbf{j} + t \mathbf{k} \), with \(t > 0\). Here, \( t^{-2} \sin t \) and \( t^{-2} \cos t \) depict diminishing circular motion in the \(xy\)-plane while amplitude shrinks rapidly. The \(t\) component adds linear upward motion, contributing to a helix that spirals inward.

Step 6: Analyze component functions in (f)

In (f): \( \mathbf{r}(t) = t^{2} \sin (\ln t) \mathbf{i} + \ln t \mathbf{j} + t^{2} \cos (\ln t) \mathbf{k} \), \( t>1 \), the terms \(t^{2} \sin (\ln t) \) and \(t^{2} \cos (\ln t) \) create expanding spirals as \( t \) grows, influenced by \(\ln t\) in \(\mathbf{j}\). This results in a helical path expanding outwards at an exponential rate.

Key Concepts

Helical Motion

Helical motion is a fascinating concept in vector calculus where an object's path follows a helix or spiral. Imagine the path of a spring or a spiral staircase. It combines two types of motion simultaneously: circular motion and linear motion.

• The circular motion occurs in a plane, often the xy-plane.
• The linear motion usually progresses along an axis, like the z-axis.

In mathematical terms, helical motion can be described using vector functions with components that create circular traces in one plane while linearly progressing in another direction. For example, when we describe a vector function such as \(\sin t \mathbf{i} + \cos t \mathbf{j} + t \mathbf{k}\), the first two components, \(\sin t \) and \(\cos t \), suggest circular movement in the xy-plane. Meanwhile, the last component, \(t \mathbf{k}\), indicates a linear path along the z-axis, forming a 3D helical motion. Helical paths are common in helical springs and DNA structures, showcasing the elegance and functionality of this geometry.

Vector Functions

Vector functions are a crucial tool in mathematics, particularly when studying paths and motions in three-dimensional space. Essentially, a vector function assigns a vector to each point on a curve, which typically varies with respect to time. This is why you often see vector functions represented as \(\mathbf{r}(t)\).

• These functions have components, such as \(\sin t \mathbf{i}\), \(\cos t \mathbf{j}\), and \(t \mathbf{k}\).
• Each component function often corresponds to a direction in space.
• For instance, the x-component, y-component, and z-component respectively.

The beauty of vector functions lies in their ability to describe complex motions elegantly. For instance, in the helical motion described earlier, the components \(\sin t\) and \(\cos t\) not only map circular motion but can also be used to compute position, velocity, and acceleration at any time \(t\). Vector functions are especially powerful in physics and engineering, where understanding the motion of particles in space is critical.

Parametric Equations

Parametric equations are widely used in vector calculus to represent curves and surfaces. Unlike standard equations, which relate \(x\) and \(y\) directly, parametric equations use one or more parameters—usually time \(t\)—to express the coordinates of points on a curve.

• For example, a parametric equation like \(x = \sin t\), \(y = \cos t\), describes a circle as \(t\) ranges over 0 to \(2\pi\).
• This representation is versatile, allowing us to represent motion along curves.
• In the helical examples, parametric equations intuitively express the motion by capturing its progression through time.

The elegance of parametric equations lies in their ability to represent three-dimensional motions and complex curves that would be hard to describe using traditional Cartesian coordinates. They are used widely in computer graphics, physics simulations, and engineering to build dynamic models and analyze movement.

Circular Motion

Circular motion is a fundamental concept in physics and mathematics, describing the motion that repeats in a circular path. This movement occurs when an object moves around a fixed point at a constant distance, forming a circle. In vector calculus, circular motion is often illustrated by functions like \(\sin t\) and \(\cos t\), representing a perfect circle in the \(xy\)-plane.

• Circular paths are uniform, meaning objects can move at a constant speed along the circle.
• For example, the motion of a planet orbiting a star can be described through circular motion.
• It's the axis of rotation that defines the plane in which circular motion occurs.

Circular motion components are crucial in creating helical paths, as seen in our vector function examples. They provide the necessary periodic motion in one plane, paired with linear progression in another axis, which together, form helices. Circular motion principles apply to various systems in everyday life, including gears, turbines, and even the wheels of a car. Understanding circular motion helps unravel these more complex patterns and applications.