Question

Derive the computational formula for curvature using the following steps. a. Use the tangential and normal components of the acceleration to show that \(\left.\mathbf{v} \times \mathbf{a}=\kappa|\mathbf{v}|^{3} \mathbf{B} . \text { (Note that } \mathbf{T} \times \mathbf{T}=\mathbf{0} .\right)\) b. Solve the equation in part (a) for \(\kappa\) and conclude that \(\kappa=\frac{|\mathbf{v} \times \mathbf{a}|}{\left|\mathbf{v}^{3}\right|},\) as shown in the text.

Short answer

Question: Calculate the curvature (\(\kappa\)) using the given velocity vector (\(\mathbf{v}\)) and the acceleration vector (\(\mathbf{a}\)). Answer: The curvature \(\kappa\) can be calculated using the formula \(\kappa=\frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|^{3}}\).

Step-by-step solution

Finding the tangential and normal components of acceleration

In this step, let's find the tangential and normal components of acceleration given the velocity vector \(\mathbf{v}\), the acceleration vector \(\mathbf{a}\), and the binormal vector \(\mathbf{B}\) using the given condition \(\mathbf{T}\times\mathbf{T}=\mathbf{0}\). First, recall that: 1. The curvature \(\kappa\) is a measure of how fast the tangent \(\mathbf{T}\) changes in a curve. The higher the curvature, the more the curve bends. 2. \(\mathbf{a} = a_T\mathbf{T} + a_N\mathbf{N}\) Now, taking the cross product of the velocity vector \(\mathbf{v}\) and the acceleration vector \(\mathbf{a}\), we get: \(\mathbf{v}\times\mathbf{a}=\mathbf{v}\times(a_T\mathbf{T}+a_N\mathbf{N})\) Applying the distributive property for cross products: \(\mathbf{v}\times\mathbf{a}=(\mathbf{v}\times a_T\mathbf{T})+(\mathbf{v}\times a_N\mathbf{N})\) Since the cross product, \(\mathbf{T}\times\mathbf{T}=\mathbf{0}\) and \(a_T\mathbf{T}\) is in the same direction as \(\mathbf{T}\), we get: \(\mathbf{v}\times\mathbf{a}=(\mathbf{v}\times a_N\mathbf{N})\)

Calculating the magnitude of the cross product

To find the curvature \(\kappa\), as we know that \(\kappa = \frac{|\mathbf{a_N}|}{|\mathbf{v}|^2}\) Using the given result, \(\mathbf{v} \times \mathbf{a}=\kappa|\mathbf{v}|^{3} \mathbf{B}\), first compute the magnitude of the cross product: \(|\mathbf{v} \times \mathbf{a}|=\kappa|\mathbf{v}|^{3} |\mathbf{B}|\) Since the magnitude of \(\mathbf{B}\) is 1, we have: \(|\mathbf{v} \times \mathbf{a}|=\kappa|\mathbf{v}|^{3}\)

Solving for curvature \(\kappa\)

Now, we need to solve the equation for \(\kappa\). To do this, divide both sides of the equation by \(|\mathbf{v}|^3\): \(\kappa=\frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|^{3}}\) This is the desired computational formula for curvature.

Key concepts

Tangential and Normal Components

In the realm of vector calculus, understanding the dynamics of how objects move involves knowing both the tangential and normal components of acceleration. Simply put, when an object moves along a curved path, its acceleration can be split into two parts:

• The **Tangential Component** aligns with the direction of velocity. It is responsible for speeding up or slowing down the object along the curve.
• The **Normal Component** acts perpendicular to the velocity. It changes the direction of the object, causing the path to curve.

To calculate these, we use the acceleration vector \(\mathbf{a}\). This vector is expressed as \(a_T\mathbf{T} + a_N\mathbf{N}\), where \(a_T\) is the tangential component and \(a_N\) is the normal component. Vector \(\mathbf{T}\) represents the tangent vector, and \(\mathbf{N}\) represents the normal vector.

Cross Product

The cross product is a fundamental concept when dealing with vectors, helping us find a vector perpendicular to two given vectors in three-dimensional space. In the context of motion analysis, taking the cross product of the velocity \(\mathbf{v}\) and acceleration \(\mathbf{a}\) vectors plays a crucial role:

• It helps us explore how these vectors interact, particularly in rotational motion contexts.
• The result of the cross product between \(\mathbf{v}\) and \(\mathbf{a}\), written as \(\mathbf{v} \times \mathbf{a}\), helps find the relationship between velocity change and path curvature.

The significance of the cross product shows up in calculating the curvature, denoted as \(\kappa\). By exploring this relationship, we learn how tightly a curve bends, which is fundamental in both physics and engineering.

Velocity and Acceleration Vectors

Velocity and acceleration vectors are essential in understanding how an object moves. Let's delve into what these vectors signify in motion analysis:

• The **Velocity Vector** \(\mathbf{v}\) defines the speed and direction of motion. It constantly changes if the object is on a curved path.
• The **Acceleration Vector** \(\mathbf{a}\) indicates how the velocity changes over time. It can cause changes in speed, direction, or both.

Understanding these vectors' roles and relationships is crucial when breaking down the components of motion, especially on a curved path. By studying them, we can discern the detailed nature of the object's path, determining factors like how quickly the direction is changing based on the calculated curvature, \(\kappa\). This understanding lays the foundation to measure and predict dynamic behavior on curves.

Magnitude Calculation

To fully understand vectors, particularly in physics and engineering contexts, being able to calculate their magnitude is key. The magnitude tells us the size or length of the vector, a critical step in many analyses:

• For the **velocity**, the magnitude represents speed. It’s derived from the magnitude of \(\mathbf{v}\), calculated as \(|\mathbf{v}|\).
• With the cross product, calculating the magnitude \(|\mathbf{v} \times \mathbf{a}|\) is vital in determining the curvature \(\kappa\). This step involves understanding how much the path deviates from a straight line on a curve.

The magnitude gives insightful details about the motion, helping us quantify the object’s dynamic state. Applying this concept to equations like \(\kappa = \frac{||\mathbf{v} \times \mathbf{a}||}{|\mathbf{v}|^3}\) reveals the curve's nature, aiding calculations and predictions regarding motion on a path.