Problem 261 - 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 \(\mathbf{F}=\omega \times \mathbf{r}\). Express \(\mathbf{F}\) in terms of \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\) vectors.

Step-by-Step Solution

Answer

\( \mathbf{F} = (bz - cy)\mathbf{i} - (az - cx)\mathbf{j} + (ay - bx)\mathbf{k} \)

Step 1: Identify the Cross Product Components

We need to express the vector field \( \mathbf{F} = \omega \times \mathbf{r} \). To do this, let's identify \( \omega \) and \( \mathbf{r} \) from the problem statement:- \( \omega = \langle a, b, c \rangle = a \mathbf{i} + b \mathbf{j} + c \mathbf{k} \)- \( \mathbf{r} = \langle x, y, z \rangle = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \)

Step 2: Set Up the Cross Product Formula

The cross product \( \mathbf{F} = \omega \times \mathbf{r} \) for vectors can be determined using the determinant of a 3x3 matrix:\[\mathbf{F} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \ a & b & c \ x & y & z \end{vmatrix} \]

Step 3: Calculate the Determinant

To calculate \( \mathbf{F} \), evaluate the determinant set in Step 2:\[\mathbf{F} = \mathbf{i} (bz - cy) - \mathbf{j} (az - cx) + \mathbf{k} (ay - bx)\]This expands:\[\mathbf{F} = (bz - cy)\mathbf{i} - (az - cx)\mathbf{j} + (ay - bx)\mathbf{k}\]

Step 4: Express the Result

Thus, the velocity vector \( \mathbf{F} \) at point \( P \) is expressed as:\[ \mathbf{F} = (bz - cy)\mathbf{i} - (az - cx)\mathbf{j} + (ay - bx)\mathbf{k} \]

Key Concepts

Angular Velocity

Angular velocity is a concept that describes how quickly a rigid body rotates. It is a vector quantity, which means it has both a magnitude and a direction. The direction of the angular velocity vector indicates the axis about which the body rotates. In the given problem, the angular velocity \( \omega = \langle a, b, c \rangle \) suggests that the rigid body is rotating with a constant speed with respect to time around a particular axis in three-dimensional space.

• Magnitude: The magnitude of the angular velocity vector tells us how fast the rotation occurs in terms of radians per second.
• Direction: Given by the components \( a \), \( b \), and \( c \), this vector is along the axis of rotation. For example, if the rotation is around the \( x \)-axis, the vector would primarily lie along this axis.

This concept helps in determining the rotational effect on any point within the rigid body, such as point \( P \) in our exercise. Understanding angular velocity is crucial for analyzing the behavior of rotating objects in mechanics.

Cross Product

The cross product, denoted by \( \times \), is an operation that takes two vectors in three-dimensional space and produces another vector perpendicular to both. This makes it a very important tool in physics, particularly in dealing with rotational motion.

• Resultant Vector: The magnitude of the resultant vector from a cross product is proportional to the sine of the angle between the two original vectors and the magnitudes of those vectors.
• Application in Rotation: In the context of our exercise, the cross product \( \omega \times \mathbf{r} \) tells us about the velocity vector field \( \mathbf{F} \).

The cross product is set up using a 3x3 determinant matrix involving the unit vectors \( \mathbf{i}, \mathbf{j}, \) and \( \mathbf{k} \). By evaluating this determinant, we obtain the expression for the velocity vector \( \mathbf{F} = (bz - cy)\mathbf{i} - (az - cx)\mathbf{j} + (ay - bx)\mathbf{k} \). This expression shows how the components of \( \omega \) and \( \mathbf{r} \) combine to create the resultant vector. Understanding the cross product is critical for determining not only the direction but also the significance of various physical phenomena in three dimensions.

Vector Field

A vector field provides a comprehensive way to model and understand how vectors change across space. Each point in the vector field is associated with a unique vector, providing a snapshot of a particular property such as velocity or force at that point.

• Defining Vector Field: For the problem at hand, the vector field \( \mathbf{F} = \omega \times \mathbf{r} \) represents the velocity field surrounding our rotating rigid body.
• Components of the Vector Field: In this setup, the field shows how the velocity of a point changes as a function of its position \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \).

The purpose of this field is to give us a full picture of motion dynamics at any point in the rigid body. This is useful in many areas of physics and engineering, where understanding such behavior helps in design and analysis of rotating objects like wheels, disks, and even orbital movements. Mastery of vector fields leads to a deeper comprehension of how an object's components move in unison.