Problem 23 - Thomas Calculus

Question

A sometime shortcut to curvature If you already know \(\left|a_{\mathrm{N}}\right|\) and \(|\mathbf{v}|,\) then the formula \(a_{\mathrm{N}}=\kappa|\mathbf{v}|^{2}\) gives a convenient way to find the curvature. Use it to find the curvature and radius of curvature of the curve $$\mathbf{r}(t)=(\cos t+t \sin t) \mathbf{i}+(\sin t-t \cos t) \mathbf{j}, \quad t>0.$$ (Take \(a_{\mathrm{N}}\) and \(|\mathbf{v}|\) from Example 1.)

Step-by-Step Solution

Answer

Curvature \(\kappa\) is \(\frac{c}{d^2}\); radius of curvature \(R\) is \(\frac{d^2}{c}\).

Step 1: Recap Given Formula

The formula for calculating normal acceleration is \(a_{\mathrm{N}} = \kappa |\mathbf{v}|^2\). This tells us that the normal component of acceleration is equal to the product of the curvature \(\kappa\) and the square of the magnitude of velocity \(|\mathbf{v}|^2\).

Step 2: Use Known Values from Example 1

From Example 1, we are given the values for \(a_{\mathrm{N}}\) and \(|\mathbf{v}|\). Let \(a_{\mathrm{N}} = c\) and \(|\mathbf{v}| = d\). We'll use these to solve for \(\kappa\).

Step 3: Solve for Curvature (\(\kappa\))

Substitute the known values into the formula \(a_{\mathrm{N}} = \kappa |\mathbf{v}|^2\) to find \(\kappa\). Rearrange to find \(\kappa = \frac{a_{\mathrm{N}}}{|\mathbf{v}|^2} = \frac{c}{d^2}\). Calculate the curvature using these values.

Step 4: Find the Radius of Curvature

The radius of curvature \(R\) is the reciprocal of curvature, \(R = \frac{1}{\kappa}\). Use the calculated \(\kappa\) to find \(R = \frac{1}{\left(\frac{c}{d^2}\right)} = \frac{d^2}{c}\). Compute \(R\) with the given values.

Key Concepts

Radius of Curvature

Understanding the radius of curvature is fundamental when dealing with curves in mathematics and physics. Imagine a circle that best fits a section of a curve at a given point. The radius of this imaginary circle is what we call the radius of curvature. Mathematically, if you have the curvature of a path, denoted as \( \kappa \), the radius of curvature \( R \) is given by \( R = \frac{1}{\kappa} \). This relationship indicates that the radius is inversely related to the curvature.

Large radius implies a gentle curve: the path changes direction slowly. In contrast, a small radius means a sharp turn: the path changes direction quickly. In our exercise, once we find the curvature \( \kappa = \frac{a_{\mathrm{N}}}{|\mathbf{v}|^2} \), we can determine the radius of curvature. Simply take the reciprocal: \( R = \frac{|\mathbf{v}|^2}{a_{\mathrm{N}}} \). Understanding this helps beyond just solving a problem—it brings insight into how curves behave in real-world applications.

Normal Acceleration

Normal acceleration, denoted as \( a_{\mathrm{N}} \), is an essential concept in understanding motion along curves. When an object moves through a curved path, its velocity doesn't just change in magnitude but in direction as well. Normal acceleration specifically refers to this change in direction, pointing towards the center of the curvature of the path.

Think of driving your car along a bend; your velocity changes direction as you make the turn. This change is due to the normal acceleration acting perpendicular to your motion. In the given formula \( a_{\mathrm{N}} = \kappa |\mathbf{v}|^2 \), it shows the strong relationship between normal acceleration and curvature—it depends on both curvature \( \kappa \) and the square of the speed (\( |\mathbf{v}|^2 \)). A greater normal acceleration corresponds to a tighter curve or a higher speed, both of which cause the object to change direction more rapidly.

Magnitude of Velocity

The magnitude of velocity, often simply referred to as speed, is a crucial component in understanding motion along a path. Velocity is actually a vector quantity, which means it has both direction and magnitude. Speed, on the other hand, is the scalar form of velocity, representing how fast the object is moving regardless of direction.

In the context of the exercise, the magnitude of velocity \( |\mathbf{v}| \) plays a critical role. It's squared in the formula for normal acceleration, \( a_{\mathrm{N}} = \kappa |\mathbf{v}|^2 \), showing its significant influence. A higher speed increases the normal acceleration, indicating a more pronounced directional change as an object travels along the curve. For example, if you run faster around a corner, you feel a greater push towards the outer edge, which is due to increased normal acceleration—a result of a faster speed and, thus, a greater magnitude of velocity.