Question

Consider the rotational velocity field \(\mathbf{v}=\langle-2 y, 2 z, 0\rangle\) a. If a paddle wheel is placed in the \(x y\) -plane with its axis normal to this plane, what is its angular speed? b. If a paddle wheel is placed in the \(x z\) -plane with its axis normal to this plane, what is its angular speed? c. If a paddle wheel is placed in the \(y z\) -plane with its axis normal to this plane, what is its angular speed?

Short answer

Question: Determine the angular speed of a paddle wheel placed in a rotational velocity field v = <-2y, 2z, 0> in the xy-plane, xz-plane, and yz-plane. Answer: a. The angular speed of the paddle wheel in the xy-plane is 0. b. The angular speed of the paddle wheel in the xz-plane is 2z. c. The angular speed of the paddle wheel in the yz-plane is -2y.

Step-by-step solution

1. Find the normal unit vector of xy-plane.

For the xy-plane, the normal unit vector would be along the z-axis, which is \(\hat{n}=\langle0,0,1\rangle\).

2. Calculate the angular speed of the paddle wheel in xy-plane.

To find the angular speed, calculate the dot product of the rotational velocity field vector and the normal unit vector: \(\omega_{xy}=\mathbf{v}\cdot\hat{n}=\langle-2y,2z,0\rangle\cdot\langle0,0,1\rangle=0\). The angular speed of the paddle wheel in the xy-plane is 0.

3. Find the normal unit vector of xz-plane.

For the xz-plane, the normal unit vector would be along the y-axis, which is \(\hat{n}=\langle0,1,0\rangle\).

4. Calculate the angular speed of the paddle wheel in xz-plane.

Calculate the dot product of the rotational velocity field vector and the normal unit vector: \(\omega_{xz}=\mathbf{v}\cdot\hat{n}=\langle-2y,2z,0\rangle\cdot\langle0,1,0\rangle=2z\). The angular speed of the paddle wheel in the xz-plane is \(2z\).

5. Find the normal unit vector of yz-plane.

For the yz-plane, the normal unit vector would be along the x-axis, which is \(\hat{n}=\langle1,0,0\rangle\).

6. Calculate the angular speed of the paddle wheel in yz-plane.

Calculate the dot product of the rotational velocity field vector and the normal unit vector: \(\omega_{yz}=\mathbf{v}\cdot\hat{n}=\langle-2y,2z,0\rangle\cdot\langle1,0,0\rangle=-2y\). The angular speed of the paddle wheel in the yz-plane is \(-2y\). In conclusion: a. The angular speed of the paddle wheel in the xy-plane is 0. b. The angular speed of the paddle wheel in the xz-plane is \(2z\). c. The angular speed of the paddle wheel in the yz-plane is \(-2y\).

Key concepts

Angular Speed

Angular speed is a measure of how quickly an object rotates or revolves about a particular point or axis. In the context of motion in a plane, such as a rotational velocity field, angular speed pertains to the rate at which the orientation of an object changes with respect to time. It is usually denoted by the symbol \( \omega \) and is expressed in radians per second (rad/s).
For a paddle wheel placed in a plane with its axis normal to that plane, the angular speed is the component of the rotational velocity that's aligned with the axis of the paddle wheel. The wheel will spin at a rate that is directly related to the magnitude of this component. To calculate this, one often uses the vector dot product between the velocity field vector and a normal unit vector to the plane. In cases where this dot product yields zero or a value related to the variables of the velocity field, it signifies that the angular speed is either zero or a function of the position within the field.

Vector Dot Product

The vector dot product, also known as the scalar product, is an operation that takes two vectors and returns a single scalar quantity. This scalar is a reflection of the magnitude of the components of one vector along the direction of another. The dot product is calculated by multiplying corresponding components of the vectors and then summing those products.
Mathematically, if we have two vectors \( \mathbf{A} = \langle a_1, a_2, a_3 \rangle \) and \( \mathbf{B} = \langle b_1, b_2, b_3 \rangle \) in three-dimensional space, their dot product is \( \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \). When the dot product equals zero, it indicates that the vectors are orthogonal, meaning they are at a right angle to one another, which is a critical condition for determining angular speed in certain planes as demonstrated in the exercise we are discussing. The vector dot product is essential for calculating quantities like work and kinetic energy in physics, as well as for determining the angle between vectors in mathematics and engineering.

Normal Unit Vector

A normal unit vector is a vector of length one (unit length) that is perpendicular to a given surface or plane. It is commonly used to represent directions in space. For a two-dimensional plane, such as the xy-plane, there is one unique normal direction, which can be captured by a normal unit vector. In three-dimensional space, the normal unit vector often represents the orientation of a plane or a surface.
When you are working with a plane defined by a pair of axes, like the xy-plane, the normal unit vector is simple to deduce—it points along the axis perpendicular to that plane. For the xy-plane, that's the z-axis, resulting in a normal unit vector \( \hat{n} = \langle 0, 0, 1 \rangle \) as seen in the provided solution. This concept is crucial for problems involving angular speed and rotational velocity fields because the paddle wheel's spin will relate to how the rotational velocity corresponds to this normal direction. Calculating the vector dot product of the velocity field with a plane's normal unit vector gives the rate of rotation, or angular speed, around that normal axis.

Velocity Field in Calculus

In calculus, specifically in the field of vector calculus, a velocity field is a spatial representation of the velocity of particles at various points in space. It assigns a vector to every point in space which indicates the velocity of a particle if it were at that point. Velocity fields are used to model fluid flow, electromagnetism, and other systems where the motion of particles is of interest.
A velocity field is expressed as a vector function of space, \( \mathbf{v}(x, y, z) \), which outputs a velocity vector for each point in space \( (x, y, z) \). In the context of the exercise at hand, we deal with a rotational velocity field, which characterizes a rotating flow, like that of a vortex or whirlpool. The paddle wheel's angular speed in such a field is determined by how the rotational flow 'impacts' or 'intersects' with the wheel's axis. By projecting the rotational velocity vector onto the normal vector of the plane of interest, as done previously, you effectively measure the contribution of the flow to the wheel's rotation in that particular orientation.