Equation of Rotational Motion

# Equation of Rotational Motion

## Introduction:

Everywhere in the universe, there is rotational motion. Rotational motion is demonstrated by the movement of electrons around an atom and the movement of the moon around the earth. When an object exhibits rotational motion, it cannot be treated as a particle because different parts of the object move at different velocities and accelerations. As a result, the object must be regarded as a particle system. The dynamics of rotational motion are identical to those of linear or translational motion. Many of the equations for rotating object mechanics are similar to linear motion equations. Only rigid bodies are considered in rotational motion. A rigid body is a mass-bearing object with a rigid shape.

Fill Out the Form for Expert Academic Guidance!

+91

Live ClassesBooksTest SeriesSelf Learning

Verify OTP Code (required)

## Overview:

Physics describes the laws and equations that govern nature and natural phenomena. The study of motion is a major focus of physics. Objects rotate around an axis. The particles and the mass center do not all move in the same way. All of the particles in the body move in the same direction. By definition, we must investigate how the various particles of a rigid body move when the body rotates.

When all parts of a rigid body move parallel to a fixed plane, the object is said to be in plane motion. There are two types of plane motion, which are as follows Pure rotational motion: In this motion, the rigid body rotates about a fixed axis that is perpendicular to a fixed plane. To put it another way, the axis is fixed and does not move or change direction in relation to an inertial frame of reference. The general plane motion is as follows: The motion, in this case, can be thought of as a combination of pure translational motion parallel to a fixed plane and pure rotational motion about an axis perpendicular to that plane

### Equation of rotational motion:

Kinematics is the study of motion. The rotational motion kinematics describes the relationships between rotation angle, angular velocity, angular acceleration, and time. Let us begin by determining an equation relating, and t. To find this equation, consider a well-known kinematic equation for translational, or straight-line, motion:

v=v 0+at

It is worth noting that in rotational motion, a = at, and we will now use the symbol a for tangential or linear acceleration.

We assume an is constant, as in linear kinematics, which implies that angular acceleration is also constant,

because a = r.

Let us now plug v = rω and a = rω into the linear equation above:

r ω= r ω 0+ r at

The radius r cancels in the equation, yielding

ω= ω 0+at

### Rotational Kinematics:

In rotational kinematics, we will look at the relationship between rotational kinematical parameters. We’ll go over the angular equivalents of the linear quantities: position, displacement, velocity, and acceleration, with which we’ve already dealt in a circular motion.

 Linear Kinematic Parameters Angular Kinematic Parameters Position s Angular position θ Displacement s=s 1– s 2 Angular displacement =θ 1– θ2 Average velocity v avg= Δ s / Δ t Average angular velocity ωavg= Δ θ / Δ t Instantaneous velocity V ins limit Δ t→0 Δ s/Δ t=d s / d t Instantaneous angular velocity ω ins limit Δ θ/Δ t=dθ / d t

We have discussed motion in a circle at a constant speed and, thus, constant angular velocity in the section on uniform circular motion. However, angular velocity is not always constant—rotational motion can speed up, slow down, or reverse directions. When a spinning skater pulls in her arms, a child pushes a merry-go-round to make it rotate, or a CD slows to a halt when turned off, the angular velocity is not constant. Because the angular velocity changes in all of these cases, angular acceleration occurs. The greater the angular acceleration, the faster the change occurs. The rate of change of angular velocity is defined as angular acceleration. Angular acceleration is expressed as an equation.

a=Δ ωt / Δt

where Δω is the change in angular velocity and t is the change in time.

If ω increases, then α is positive. If decreases, then it is negative.

Keep in that, by convention, counterclockwise is the positive direction and clockwise is the negative direction.

### Axis of rotation:

The figure depicts a rigid body of any shape rotating about a fixed axis (the axis that does not move) known as the axis of rotation or rotation axis.

When a rigid body rotates, there is a line that all the parts revolve around. The parts farther away from that line move faster because they travel in a larger circle around that line. Parts closer to the line move more slowly as a result of following smaller circles. Points that are directly on the line do not travel at all. They simply turn in place. This line is the rigid body’s axis of rotation. If the rigid body is a wheel, the axis is the axle located in the center of the wheel.

Spin rotation of a rigid object can occur in conjunction with translational motion. We’ll save the description of combined translational and rotational motion for later. In this module, we’ll look at how to describe pure translational motion. Pure rotational motion can be extremely complex, and some cases are beyond the scope of any introductory physics course.

To simplify the concepts of angular motion, we will impose the following constraints:

The rigid body revolves around a fixed axis.

The rotation occurs in the plane that contains the object, such as the x y plane in the figure below.

The rotation axis is perpendicular to the plane in which the object is contained, denoted by the z-axis in the figures below.

### Types of Motion involving Rotation are as follows:

1. Rotation about a fixed axis
2. Rolling
3. Rotation about an axis in the rotation

### Rotation about a fixed axis:

Examples include the rotation of a ceiling fan, the opening and closing of a door, the rotation of our planet, and the rotation of the hour and minute hands in analog clocks.

As the rigid body rotates, each point on the axis remains stationary, while each point not on the axis moves in a circular path around the axis.

When a rigid body rotates around a fixed axis, all of its points experience the same linear displacement. A rigid body rotates about a fixed axis, all of its points have the same angular speed. When a rigid body rotates about a fixed axis, all of its points have the same tangential speed.

### Rolling:

This category includes activities such as rolling.

The rolling of wheels and wheel-like objects is arguably the most important application of rotational physics, as our world is now filled with automobiles and other rolling vehicles. A body’s motion is a combination of translational and rotational motion of a round-shaped body on a surface. When a body is in a rolling motion, each particle has two velocities: one due to rotational motion and the other due to translational motion (of the center of mass), and the resulting effect is the vector sum of both velocities at all particles.

Question 1:What is rotational motion? Give an example.

Answer: A rotational motion is one in which the body moves in a circular motion.

An example is the car wheel.

Question 2:What is the reason for rotational motion?

The torque, also known as rotational analog force, is a cause of rotational motion. When torque is applied to the particle’s system about its axis, it causes a twist, which is the cause of rotational motion.

Question 3: Is circular motion the same as rotational motion? Explain

Answer: Circular motion is defined as the movement of the body around a fixed point. A fixed point exists outside the body in this case. The circular motion is due to centripetal force in this case. The rotational motion is the fixed point in rotational motion is located within the body. The torque acting on the particle system causes rotational motion

## Related content

 Degloved Face Injury | Causes, Types, Treatment, Recovery PM Yashasvi Scholarship 2023, Dates, Registration, Eligibility, Benefits, Documents List of Home Ministers of India – Who is the Home Minister of India? Difference Between Mass and Weight Differences & Comparisons Articles in Physics YouTube Downloader – Smart Hacks, Tips, Tricks and Tutorials JEE Main Reduced Syllabus 2024 with Tips to Ace the Exam Best Smart Strategies to Prepare for JEE Main 2024 in 3 Months 5 Hacks to Score High in JEE Main 2024 Maths Exam 7 Intelligent Tips to Score 90+ in CBSE Class 12 Physics  +91

Live ClassesBooksTest SeriesSelf Learning

Verify OTP Code (required)