Problem 263 - Calculus Volume 3

Question

Consider a rigid body that is rotating about the \(x\) -axis counterclockwise with constant angular velocity \(\omega=\langle a, b, c\rangle .\) If \(P\) is a point in the body located at \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k},\) the velocity at \(P\) is given by vector field \(\mathbf{F}=\omega \times \mathbf{r}\). Find \(\operatorname{curl} \mathbf{F}\)

Step-by-Step Solution

Answer

The curl of \( \mathbf{F} \) is \( 2a \mathbf{i} \).

Step 1: Understand the Given Information

We are given a velocity field expressed as \( \mathbf{F} = \omega \times \mathbf{r} \), where \( \omega = \langle a, b, c \rangle \) is the angular velocity vector and \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \) is the position vector of the point \( P \). We need to find the curl of the vector field \( \mathbf{F} \).

Step 2: Recall the Curl Formula

The curl of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by the determinant:\[abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix}\]

Step 3: Express \( \mathbf{F} \, \text{in Components} \) as a Cross Product

Calculate \( \mathbf{F} = \omega \times \mathbf{r} \):\[\begin{align*}\mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a & b & c \ x & y & z \end{vmatrix} = \left(bz - cy\right)\mathbf{i} + \left(cx - az\right)\mathbf{j} + \left(ay - bx\right)\mathbf{k}\end{align*}\]

Step 4: Compute \( \operatorname{curl} \mathbf{F} \)

Substitute the components of \( \mathbf{F} \) into the curl formula from Step 2:\[\begin{align*}\operatorname{curl} \mathbf{F} &= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ bz - cy & cx - az & ay - bx \end{vmatrix} \&= \mathbf{i} \left(\frac{\partial}{\partial y}(ay - bx) - \frac{\partial}{\partial z}(cx - az) \right) - \mathbf{j} \left(\frac{\partial}{\partial x}(ay - bx) - \frac{\partial}{\partial z}(bz - cy) \right) \& + \mathbf{k} \left(\frac{\partial}{\partial x}(cx - az) - \frac{\partial}{\partial y}(bz - cy) \right)\end{align*}\]

Step 5: Evaluate Partial Derivatives

Evaluate the partial derivatives:\[\begin{align*}\mathbf{i}&: \frac{\partial}{\partial y}(ay - bx) = a, \, \frac{\partial}{\partial z}(cx - az) = -a \\mathbf{j}&: \frac{\partial}{\partial x}(ay - bx) = -b, \, \frac{\partial}{\partial z}(bz - cy) = -c \\mathbf{k}&: \frac{\partial}{\partial x}(cx - az) = c, \, \frac{\partial}{\partial y}(bz - cy) = b\end{align*}\]

Step 6: Complete the Curl Calculation

Substitute the evaluated derivatives into the expressions found in Step 4:\[\begin{align*}\operatorname{curl} \mathbf{F} &= \mathbf{i}(a + a) - \mathbf{j}(-b + c) + \mathbf{k}(c - b) \&= 2a \mathbf{i} + (b - c) \mathbf{j} + (c - b) \mathbf{k}\end{align*}\]

Step 7: Final Step: Simplify the Result

Since the terms \( b - c \) and \( c - b \) differ only in sign, the simplified result of the curl is:\[\operatorname{curl} \mathbf{F} = 2a \mathbf{i}\]

Key Concepts

Cross Product

The cross product is a way to produce a vector that is perpendicular to two given vectors in a three-dimensional space. It results in another vector, which can be used in physics and engineering to find quantities like torque or angular momentum.
The formula for the cross product of two vectors \( \mathbf{A} = A_x \mathbf{i} + A_y \mathbf{j} + A_z \mathbf{k} \) and \( \mathbf{B} = B_x \mathbf{i} + B_y \mathbf{j} + B_z \mathbf{k} \) can be expressed as a determinant:

• \( \mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \end{vmatrix} \)

By solving this determinant, you obtain the components of the resulting cross product vector. This cross product vector, in the context of the problem, is the velocity vector \( \mathbf{F} = \omega \times \mathbf{r} \) that describes how fast each point in a rotating body is moving due to the rotation.
Understanding this concept helps in solving for other elements like the curl in vector calculus, which can tell us how a vector field rotates around points.

Angular Velocity

Angular velocity quantifies the rotation rate of an object or how fast it spins around a fixed axis. It is a vector quantity, meaning it has both magnitude and direction. The direction is determined by the right-hand rule, and the magnitude measures how fast the rotation occurs.

• Angular velocity is commonly represented by \( \boldsymbol{\omega} \), a vector in three-dimensional space.
• For a rigid body rotating counterclockwise around the x-axis, the angular velocity vector might be expressed as \( \omega = \langle a, b, c \rangle \).

This vector addresses the concept of rotational movement and plays a crucial role in physics and engineering. In our context, it determines the rotational speed of a rigid body and provides necessary information to compute the velocity of points on that body using the cross product. Knowing the vector direction also aids in determining the rotational characteristics such as the curl, which expresses the distribution of rotational energy.

Rigid Body Rotation

Rigid body rotation involves studying objects that maintain a fixed shape while rotating about an axis. This idealization simplifies many real-world problems, where the body's shape and volume do not change.
In rigid body dynamics,

• All particles of the body move in circular paths around the axis of rotation.
• The angular velocity is constant across the entire body.

Consider a rigid body rotating about a fixed x-axis in physics, where every point in the body takes a circular path in a plane perpendicular to this axis. Conceptually, there are no deformations or shifts in particle positions relative to one another, maintaining the body's structure intact. Analyzing problems in this context typically involves utilizing vector concepts like the angular velocity to determine resultant fields like the velocity vector field. Understanding rigid body rotation allows one to properly calculate important aspects such as angular momentum, kinetic energy, and the effects of applied torques, feeding into broader topics such as the curl of vector fields.

Vector Field

A vector field assigns a vector to every point in space. They are essential in visualizing and analyzing numerous physical phenomena like fluid flow, electromagnetic fields, and others. Each vector of the field contains magnitude and direction, providing detailed information on how different quantities behave in space.
To understand a vector field:

• Visualize a three-dimensional space filled with vectors pointing in various directions.
• Every point in the space has a corresponding vector, indicating properties like velocity or force at that point.

In this problem, the vector field \( \mathbf{F} = \omega \times \mathbf{r} \) represents the velocity field of a rotating body. Each vector of the field gives the velocity at specific points inside a rotating body, derived through the cross product of angular velocity and position vectors. Grasping how vector fields operate sets the stage for understanding more complex operations like finding divergence and curl—tools used throughout physics and engineering to determine how fields evolve and interact.

Partial Derivatives

A partial derivative represents how a function changes as one of its variables changes, while other variables are held constant. In three-dimensional calculus, this allows us to explore how functions of multiple variables behave.
Consider a function \( f(x, y, z) \):

• The partial derivative with respect to \( x \) is denoted by \( \frac{\partial f}{\partial x} \).
• Partial derivatives help analyze multidimensional changes by isolating each variable's influence.

In vector calculus, partial derivatives are instrumental in computing curls and divergences, essential operations for analyzing vector fields. For our exercise, calculating the curl of \( \mathbf{F} \) involves determining partial derivatives of its components. Solving these derivatives provides insights into rotational characteristics of the velocity vector field generated by the rigid body's rotation. Understanding partial derivatives’ role is crucial for developing a solid foundation in multivariable calculus, aiding in learning how changes in one direction affect the overall function.