A wheel is spinning about a horizontal axis with angular speed \(140 \mathrm{rad} / \mathrm{s}\) and with its angular velocity pointing east. Find the magnitude and direction of its angular velocity after an angular acceleration of \(35 \mathrm{rad} / \mathrm{s}^{2},\) pointing \(68^{\circ}\) west of north, is applied for \(5.0 \mathrm{s}\).

Angular acceleration is fundamental to understanding how objects spin and change their rate of rotation. It's defined as the rate at which the angular velocity of an object changes with respect to time. In mathematical terms, if the angular velocity vector is represented by \( \vec{\omega} \), then the angular acceleration \( \vec{\alpha} \) is the time derivative of \( \vec{\omega} \) or \( \vec{\alpha} = \frac{d\vec{\omega}}{dt} \).

It is a vector quantity, meaning it has both a magnitude and a direction. The magnitude tells us how quickly the rotation speed is changing, while the direction indicates the axis along which the rotation rate is increasing or decreasing. The SI unit of angular acceleration is radians per second squared \( (\mathrm{rad/s^2}) \). In our exercise, the wheel initially at rest experiences an angular acceleration, which results in a change in the angular velocity over time.

In physics, vectors are used to represent quantities that have both magnitude and direction. They are essential in describing physical phenomena, especially in mechanics. A vector is often denoted by an arrow, where the length represents the magnitude, and the arrow points in the direction of the vector. Common vector quantities include displacement, velocity, acceleration, and force.

For two-dimensional vectors, we usually use the unit vectors \( \hat{i} \) and \( \hat{j} \) to represent the horizontal and vertical components, respectively. In the context of angular motion, the initial angular velocity \( \vec{\omega}_i \) and the angular acceleration \( \vec{\alpha} \) are written in terms of \( \hat{i} \) and \( \hat{j} \) to show their components in the east-west and north-south directions. This vector representation helps in calculating the resultant angular velocity by adding vector components algebraically.

Angular motion equations are analogs to linear motion equations but for rotating bodies. One of the key equations relates the final angular velocity \( \vec{\omega}_f \) to the initial angular velocity \( \vec{\omega}_i \) plus the product of angular acceleration \( \vec{\alpha} \) and time \( t \), given by \( \vec{\omega}_f = \vec{\omega}_i + \vec{\alpha} t \). This equation is crucial when analyzing problems involving rotational motion with constant angular acceleration. It allows us to calculate the new angular velocity after a certain amount of time, given the initial conditions and the angular acceleration applied, as demonstrated in the provided exercise.

By understanding and utilizing these equations, one can solve for various parameters of rotational movement such as angular displacement, final angular speed, and the total time taken for particular angular motions.

Angular displacement is the angle through which a point, line, or body has been rotated in a specified sense about a specified axis. It is a measure of the change in the angular position of a rotating object. As it's an angle, the most common unit for angular displacement is radians, though it can also be measured in degrees. Unlike angular velocity and acceleration, which are vector quantities, angular displacement is a scalar quantity as it does not depend on the direction of rotation.

Mathematically, angular displacement \( \Delta \theta \) over a time interval can be found if the angular velocity \( \vec{\omega} \) is constant, using the relation \( \Delta \theta = \vec{\omega} t \) where \( \vec{\omega} \) is the angular velocity and \( t \) is the time. In more complex scenarios with variable angular velocity, calculus methods are often employed to calculate angular displacement.