Rotational Motion

Rotational Motion Definition

Below we will define rotational motion and discuss how it is divided into different types.

Types of Rotational Motion

Rotational Motion can be divided into three types.

1. Motion about a fixed axis: Is also known as pure rotation and describes the rotation of an object around a fixed point. Some examples are the rotating of fan blades or the rotating of hands on an analog clock as both rotate about a central fixed point.
2. A combination of rotational and translational motion. This motion describes an object, whose components can rotate about a fixed point, while the object itself travels along a linear path. An example is the rolling of wheels on a car. The wheels have two velocities, one as a result of the rotating wheel and another due to the car's translational motion.
3. Rotation about an axis of rotation. This motion describes objects that rotate about an axis while also rotating around another object. An example is Earth orbiting around the sun while it also rotates about its own axis.

Rotational Motion Physics

The physics behind rotational motion is described by a concept known as kinematics. Kinematics is a field within physics that focuses on the motion of an object without referencing the forces causing the motion. Kinematics focuses on variables such as acceleration, velocity, displacement, and time which can be written in terms of linear or rotational motion. When studying rotational motion, we use the concept of rotational kinematics. Rotational kinematics refers to rotational motion and discusses the relationship between rotational motion variables.

Rotational Motion Variables

The rotational motion variables are:

1. angular velocity
2. angular acceleration
3. angular displacement
4. time

Angular Velocity, \(\omega\)

Angular velocity is the change in the angle with respect to time. Its corresponding formula is $$ \omega = \frac{\theta}{t}$$ where angular velocity is measured in radians per second, \(\mathrm{\frac{rad}{s}}\).

Angular Acceleration , \(\alpha\)

Angular acceleration is the change in angular velocity with respect to time. Its corresponding formula is $$ \alpha = \frac{\omega}{t} $$ where angular acceleration is measured in radians per second squared, \(\mathrm{\frac{rad}{s^2}}\).

Angular Displacement, \(\theta\)

Angular displacement is the product of angular velocity and time. Its corresponding formula is $$ \theta = \omega t $$ where angular displacement is measured in radians, \(\mathrm{rad}\).

Time, \(t\)

Time is time. $$ \mathrm{time} = t $$ where time is measured in seconds, \(s\).

Relationship Between Rotational Kinematics and Linear Kinematics

Before diving deeper into rotational kinematics, we must be sure to recognize and understand the relationship between kinematic variables. This can be seen when looking at the variables in the table below.

As a result, kinematic equations of motion can be written in terms of linear and rotational motion. However, it is important to understand that although equations are written in terms of different variables, they are of the same form because rotational motion is the equivalent counterpart of linear motion.

Rotational Motion Formulas

The relationship between rotational motion and rotational motion variables is expressed through three kinematic equations, each of which is missing a kinematic variable.

$$\omega=\omega_{o} + \alpha{t}$$

$$\Delta{\theta} =\omega_o{t}+\frac{1}{2}{\alpha}t$$

$$\omega^2={\omega_{o}}^2 +2{\alpha}\Delta{\theta}$$

where \(\omega\) is final angular acceleration, \(\omega_0\) is the initial angular velocity, \(\alpha\) is angular acceleration, \(t\) is time, and \( \Delta{\theta} \) is angular displacement.

Rotational Kinematics and Rotational Dynamics

As we have discussed rotational kinematics, it is also important for us to discuss rotational dynamics. Rotational dynamics deals with the motion of an object and the forces causing the object to rotate. In rotational motion, we know this force is torque.

Newton's Second Law for Rotational Motion

Below will we define torque and its corresponding mathematical formula.

Torque

In order to formulate Newton's second law in terms of rotational motion, we must first define torque.

The equation for torque can be written in the same form as Newton's second law, \(F=ma\), and is expressed as $$\tau = I \alpha$$

where \(I\) is the moment of inertia and \(\alpha\) is angular acceleration. Torque can be expressed this way as it is the rotational equivalent of force.

However, when the system is at rest, it is said to be in rotational equilibrium. Rotational equilibrium is defined as a state in which neither a system's state of motion nor its internal energy state changes with respect to time. Therefore, for a system to be at equilibrium, the sum of all forces acting on the system must be zero. In rotational motion, this means that the sum of all torques acting on a system must equal zero.

$$ \sum \tau = 0 $$

Torque and Angular Acceleration

The relationship between angular acceleration and torque is expressed when the equation, \( \tau={I}\alpha \) is rearranged to solve for angular acceleration. As a result, the equation becomes\( \alpha=\frac{\tau}{I} \). Thus, we can determine that angular acceleration is proportional to torque and inversely proportional to the moment of inertia.

Rotational Motion Examples

To solve rotational motion examples, the five rotational kinematic equations can be used. As we have defined rotational motion and discussed its relation to kinematics and linear motion, let us work through some examples to gain a better understanding of rotational motion. Note that before solving a problem, we must always remember these simple steps:

1. Read the problem and identify all variables given within the problem.
2. Determine what the problem is asking and what formulas are needed.
3. Apply the necessary formulas and solve the problem.
4. Draw a picture if necessary to provide a visual aid

Example 1

Let us apply the rotational kinematic equations to a spinning top.

Example 2

Next, we will do the same thing for a tornado.

Example 3

For our last example, we will apply the torque equation to a rotating object.