Equations of Rotational Motion

A rotational motion is defined as the movement of a point in a circular path with an axis of rotation that cannot be altered. In rotational dynamics, the causes of rotational motion are taken into account along with its attributes, while in rotational kinematics, rotational motion is evaluated without addressing its causes.

Let’s understand rotational motion at a basic level. When all of a rigid body’s particles move in a circular motion, and the centres of these circles remain constant on a specific straight line called an axis of rotation, that movement is referred to as rotational motion. Fundamentally, rotational motion refers to the motion of an object in a circular path around a fixed point.

Rotational motion is divided into two parts: pure rotational motion (rotation about a fixed axis) and combined translational and rotational motion (CTRM). Here, we will study the kinematics of rotational motion about a fixed axis.

As you know, rotational motion is similar to translational (linear) motion in many aspects. Even the terms we use in rotational motion are analogous to the terms used in translational motion. Before we discuss in detail the kinematics of rotation about a fixed axis, let’s quickly take a look at the following definitions.

Types of Motion Involving Rotation

1. Rotation around a central axis: This includes the rotation of the ceiling fan, the potter’s wheel, the opening and closing of doors, the rotation of the clock hands on the wall, and so on. When the ceiling fan rotates, the vertical rod from which it is suspended remains stationary while its blades move in a circular path.
2. Rotation around the translational axis stimulates a wide range of motion. This type of motion includes things like return motion. Consider the wheel speed of a vehicle travelling on a straight, flat road compared to the speed of a guide system moving alongside the vehicle. The wheel appears to be rotating around a stationary axle. The wheel will rotate around a moving axle with the guide system attached to the ground. As a result, the wheel’s returning speed results from the superposition of two separate motions that co-occur.
3. Rotation around a central axis: The object rotates around one axis, which revolves around another axis in this motion.

Rotational Kinematics Variables

Translational motion involves five primary variables: position, initial velocity, final velocity, acceleration, and time. Each linear variable has a corresponding rotational variable. The parameters of rotational motion are defined as follows.

Angular distance (θ)

It is the angle at which a particle rotates around a specified axis. θ (theta) is measured in radians.

Angular velocity (ω)

It is the rate of change of angular distance (θ) with respect to time. ω is measured in radians per second.

Angular acceleration (α)

It is the rate of change of angular velocity (ω) with respect to time. α is measured in radians per second squared.

Time

Time is time, irrespective of whether the particle is showing translational motion or rotational motion.

Basic Rotational Motion Equations

Let’s suppose a particle is undergoing rotational motion about a fixed axis. The particle moves from one point to another, such that the angular displacement is θ.

By definition, angular velocity (ω) is the rate of change of angular displacement. So, mathematically,

ω = dθ/dt

and, similarly,

Angular acceleration α = dω/dt

Since rotational motion parameters are analogous to linear motion variables, rotational motion equations will be similar to kinematical equations for linear motion.

Let’s recall equations for linear motion with constant acceleration:

v = v₀+ at

x = x₀ + v₀t + (1/2) at²

v² = v₀²+ 2a (x – x₀)

where, x₀ is the initial displacement, v₀ is the initial velocity, x and v are the displacement and velocity of the particle at any given time respectively, a is the acceleration, and t is the time.

The corresponding kinematic equations for rotational motion will be:

ω = ω₀+ αt

θ = θ₀ + ω₀t + (1/2) αt²

ω² = ω₀²+ 2α (θ – θ₀)

Some Other Rotational Motion Formulas

Angular displacement θ = x/r

Angular velocity ω  = v/r

Angular acceleration α = a/r

Average angular velocity ω_avg = (ω_{0 + ω)/2}

No. of rotations of a particle at any given time N = θ/2π

where x, v, a are linear motion parameters, and r is the radius of curvature of the circular path.

The following table shows the analogous relationship between translational and rotational motion.

Conclusion

In a nutshell, kinematics is the description of motion. If you understand translational motion, rotational motion will be easier for you. All you have to do is substitute all the translational motion parameters with rotational motion variables.

We have learned about the relationship between various rotational parameters, the analogous relationship between translational and rotational motion, and strategies to solve rotational kinematics problems. Below are a few questions that may help you further with the topic.