Problem 26 - Thomas Calculus

Question

The torsion of a helix In Example \(2,\) we found the torsion of the helix $$ \mathbf{r}(t)=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}+b t \mathbf{k}, \quad a, b \geq 0 $$ to be \(\tau=b /\left(a^{2}+b^{2}\right) .\) What is the largest value \(\tau\) can have for a given value of \(a ?\) Give reasons for your answer.

Step-by-Step Solution

Answer

For a given value of \(a\), the largest torsion \(\tau\) is \(\frac{1}{2a}\) at \(b = a\).

Step 1: Identify Given Information

We are given a parameterization of a helix defined by \( \mathbf{r}(t) = (a \cos t) \mathbf{i} + (a \sin t) \mathbf{j} + b t \mathbf{k} \), where \(a\) and \(b\) are non-negative constants. The torsion of this helix is \( \tau = \frac{b}{a^2 + b^2} \).

Step 2: Understand the Problem

We need to find the maximum value of the torsion \(\tau\) given a specific value of \(a\). Given that \(b \geq 0\), we want to find the maximum value of the expression \(\frac{b}{a^2 + b^2}\) with respect to \(b\).

Step 3: Use Calculus to Maximize the Torsion

Introduce a function \( f(b) = \frac{b}{a^2 + b^2} \) and differentiate it with respect to \(b\). Using the quotient rule, we find:\[ f'(b) = \frac{(a^2 + b^2)(1) - b(2b)}{(a^2 + b^2)^2} = \frac{a^2 - b^2}{(a^2 + b^2)^2}\]To find critical points, set \( f'(b) = 0 \), which gives \( a^2 - b^2 = 0 \), or \( b = a \).

Step 4: Verify Maximum Value

Substitute back \( b = a \) into the expression for torsion:\[ \tau = \frac{a}{a^2 + a^2} = \frac{a}{2a^2} = \frac{1}{2a}\]This confirms a critical point. Upon inspection, \( \frac{b}{a^2 + b^2} \) as \( b \rightarrow 0 \) gives \( \tau \rightarrow 0 \), and as \( b \rightarrow \infty \), \( \tau \rightarrow 0 \) as well, confirming that \( \frac{1}{2a} \) is indeed the maximum value.

Key Concepts

Helix Torsion

Torsion is a measure of how quickly a curve is twisting out of its plane. For the helix given by the parameterization \( \mathbf{r}(t) = (a \cos t) \mathbf{i} + (a \sin t) \mathbf{j} + b t \mathbf{k} \), the torsion describes how much this helix spirals upward as it wraps around the cylinder.
When examining a helix, the torsion \( \tau \) gives us insightful information about its spatial properties. Specifically, it tells us how sharply the helix is twisting. For our given helix, the torsion is represented by \( \tau = \frac{b}{a^2 + b^2} \).
Here, \( a \) determines the radius of the helix and \( b \) dictates the pitch, or how much it rises for each complete turn around the z-axis. Large torsion values can indicate tight spirals, whereas small torsion values suggest gentle, wide loops.

Parameterization

Parameterization is a way to represent a curve by a vector function with a parameter. In the context of our helix, parameterization allows us to express it as \( \mathbf{r}(t) = (a \cos t) \mathbf{i} + (a \sin t) \mathbf{j} + b t \mathbf{k} \).
This denotes a helix on the surface of a cylinder with radius \( a \) and height controlled by \( b \), as the variable \( t \) changes. This way of describing curves is essential in calculus because it allows us to use techniques like differentiation and integration to examine properties such as torsion, curvature, and arc length.
The parameter \( t \) can be imagined as time, and as \( t \) increases, the trace of the vector \( \mathbf{r}(t) \) draws the path of the helix in 3D space.

Critical Points

Critical points occur where the derivative of a function is zero. They are essential for identifying where maximum or minimum values occur.
In our exercise, we find critical points of \( f(b) = \frac{b}{a^2 + b^2} \), the torsion function concerning \( b \). By setting the derivative \( f'(b) = 0 \), we discover that \( b = a \) is a critical point.
Determining critical points allows us to analyze changes in the function's behavior, such as finding where the torsion reaches its maximum. These points can be thoroughly examined using tests like the second derivative test to confirm whether the critical point is indeed a maximum or minimum.

Maximum Value

Finding the maximum value involves analyzing the potential extrema that a function can attain based on its critical points and behavior at the endpoints of its domain.
For torsion \( \tau = \frac{b}{a^2 + b^2} \), we calculated the critical point at \( b = a \). Substituting \( b = a \) back into the expression for \( \tau \) gives us \( \frac{1}{2a} \), confirming it as the maximum value.
Checking the limits as \( b \to 0 \) and \( b \to \infty \), we see that \( \tau \to 0 \) in both cases, which verifies the critical point we found truly yields the highest torsion possible with given constraints. Understanding maximum values is crucial in optimizations where specific results are targeted, like maximizing the efficiency of a curved structure or path.