What is Kinematics of Rotational Motion About a Fixed Axis: Basic Concepts, Equations, Derivations & Real Life Examples

The motion of an object can be divided into two categories:

• Moving from one point to another (called linear motion)
• Rotating around a fixed axis (called rotational motion)

Rotational motion is referred to as the motion which occurs when the body rotates around a fixed axis unlike linear motion in a straight line. The object rotates around a stationary point or line called the axis of rotation. Each particle of the object moves in a circular path with axis being the centre point. How far the centre axis is from the particle will determine the radius of the circle. Some common examples include a spinning top, earth revolving around the sun, moving fan, hands of a clock, etc.

To study this topic briefly, you need to be familiar with some key terms which we have covered in the article further. Continue scrolling!

In rotational motion about a fixed axis, the motion is described using angular quantities and concepts which are explained as follows:

Angular Displacement $\mathit{\theta }$ :

Angular Displacement is referred to the angle at which the object rotates around the fixed point of axis. The formula for angular displacement is:

θ = θf - θi

Where,

θ = angle of displacement

θi = angle of initial position

θf = angle of final position

 

It is a vector quantity and is measured in radians.

Angular Velocity $\mathit{\omega }$ :

Angular Velocity is the measure of the speed of rotation of a particular body. It can be represented by the mathematical equation:

$\omega =\frac{d\theta }{dt}$

Where,

ω = angular velocity

dθ/dt = rate of change

It is a vector quantity and SI unit of angular velocity is radian per seconds.

Angular Acceleration $\left(\mathit{\alpha }\right)$ :

Angular Acceleration is referred to the rate of change in the acceleration of the object. Just as linear acceleration describes how fast the velocity of an object changes from one point to another, angular acceleration tells us how fast the object moves around its fixed point of axis. It can be denoted through the formula:

$\alpha =\frac{d\omega }{dt}$

Where,

α = angular acceleration

ω = angular velocity

t = time

The SI Unit for angular acceleration is $\mathrm{r}\mathrm{a}\mathrm{d}/{\mathrm{s}}^2$ .

Torque $\vec{\tau }$ :

Torque is considered as the rotational equivalent of force and measures the force applied on the object rotating on the axis. it is caused by external forces acting at a distance from the axis, and the moment of inertia quantifies the body's resistance to angular acceleration.

Mathematical Representation:

$\vec{\tau }=I\vec{\alpha }$

Where,

$\vec{\tau }$ = net external torque about the axis,

$I$ = moment of inertia of the body about the axis

$\vec{\alpha }$ = angular acceleration.

SI unit of torque is $\mathrm{N}\cdot \mathrm{m}$ .

There are some important formulas which need to be kept in mind while diving into the world of kinematics, known as kinematic equations. In the case of constant angular acceleration, the kinematic equations mirror those of linear motion. For such scenarios, the following equations can be used:

1. $\omega ={\omega }_0+\alpha t$
2. $\theta ={\omega }_{0}t+\frac{1}{2}\alpha t^2$
3. ${\omega }^2={\omega }_0^2+2\alpha \left(\theta \right)$

Here, ${\theta }_0$ is the initial angular displacement, ${\omega }_0$  is the initial angular velocity, and $\alpha$ is the constant angular acceleration.

Now let us have a look at the derivations of each of these equations in detail.

1^st Equation of Rotational Motion

$\omega ={\omega }_0+\alpha t$

First of all, α=dω​/dt

Integrating both the sides, we get

ω−ω0​=αt

ω=ω0​+αt

Hence, Proved!

2^nd Equation of Rotational Motion

$\theta ={\omega }_{0}t+\frac{1}{2}\alpha t^2$

According to the equation of angular velocity:

ω=dθ​/dt

On substituting ω from the first equation and integrating both sides, the resulting equations turns out to be:

dθ​/dt = ω0​+αt

θ = ω0​t+1/2​αt^2

Hence, Proved!

3^rd Equation of Rotational Motion

$\theta ={\omega }_{0}t+\frac{1}{2}\alpha t^2$

α=dω​/dt (1)

ω=dθ​/dt (2)

Now,

α=dtdω​=dθdω​⋅dtdθ​=dθdω​⋅ω

αdθ=ωdω

Integrate the above equation and get:

αθ = 1​/2ω^2−1/2​ω0^2

Now multiply both sides by 2:

2αθ=ω2−ω02​

2αθ + ω02​=ω2

Hence, Proved!

Just as we use the terms mass of an object in linear motion, moment of inertia is the terms used in scenarios involving rotational motion. Yes, it is known as the rotational equivalent of mass in linear motion and is used to calculate the resistance shown by an object while changing its motion.

For a system of particles, moment of inertia will be represented as:

$I=\sum m_i{r}_i^2$

Where,

I = Moment of Inertia

Mi = Mass of the body

Ri = Distance from the axis

For continuous bodies, it can be presented as: $I=\int r^{2}dm$

Know More About: NCERT Chapter 6: Systems of Particles and Rotational Motion

Kinematic equations are applied to numerical problems involving rotating objects like fans, pulleys, or discs etc. Apart from these, there are also numerous real world uses of this concept which have assisted in the invention and development of various modern day technologies which help in the advancement of technology. Here are a few key applications of rotational motion in the real world:

• Motors and Engines
• Automobiles
• Biomechanics
• Sports
• Gyroscopes
• Transportation
• Astronomy

Here are some important accumulated topics and their notes which will help you prepare for JEE MAINS effectively.

Units and Measurements Class 11 Notes	Mechanical Properties of Solids Class 11 Notes
Motion in a Straight Line Class 11 Notes	Mechanical Properties of Fluids Class 11 Notes
NCERT Class 11 Notes for Motion in a Plane	Thermal Properties of Matter Class 11 Notes
Laws of Motion Class 11 Notes	Thermodynamics Class 11 Notes
Work, Energy, and Power Class 11 Notes	Kinetic Theory of Gas Class 11 Notes
System of Particles and Rotational Motion Class 11 Notes	Oscillations Class 11 Notes
Gravitation Class 11 Notes	Waves Class 11 Notes

Click Here:

NCERT Class 11 Notes for PCM

NCERT Class 11 Physics Notes