Equation of rotational motion

Introduction

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 rotating kinetics, 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.

Body

Because we’re dealing with a rigid body, we can assume that each particle’s angular velocity will be the same. Rotational kinematics is the study of rotational motion without taking into account the cause of the rotation and the changes in its attributes. When the causes of rotational motion are taken into account, as well as the body’s constant and changing properties, we call it rotational dynamics. Rotational dynamics entails a quick examination of the system as well as the examination of the body’s parameters while it is in motion.

By specifying parameters such as angular displacement, angular velocity, and angular acceleration, we may analyse rotational dynamics. We must also define intrinsic qualities such as the moment of inertia, the centre of mass, and so on. Newton’s Laws of Motion will be applicable here by altering the linear coordinates to angular coordinates as we are examining the forces and motion of the body. Similarly, the equations for evenly accelerated motion can be used.

A motion of an object around a circular route in a fixed orbit is also known as rotational motion. It can alternatively be defined as a body’s motion around a fixed point in which all of its particles travel in a circular motion with the same angular velocity—for example, Earth’s rotation around its axis.

The rotation of an item around a fixed point can be done in one of two ways: clockwise or anticlockwise. Circular energy is the energy produced by this rotational motion. Torque, the moment of inertia, angular momentum, and other important words are all linked with rotational motion.

• Torque is the magnitude of force put on a body that is applied away from the point of application of force. The “turning effect” is another name for it. It’s easier to push a door open at its ends than at the hinges, for example.
• The moment of inertia is a property that influences how much torque is required to rotate an object about its axis. Rotational inertia is another name for it.
• Angular momentum is the product of angular velocity and moment of inertia, and it determines the body’s rotational quantity.

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 ωavg = (ω0 + 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.

Table showing the analogous relationship between translational and rotational motion

Conclusion:

In a nutshell, kinematics is the description of motion. If you know translational motion, the rotational motion will be a piece of cake for you. All you have to do is substitute all the translational motion parameters with rotational motion variables. You can read more articles on translation motion on the Unacademy website.

So far, 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 more questions that may help you with the topic.