What is Dynamics of Rotational Motion About a Fixed Axis: Definition, Formulas, Derivations, Newton's Law & Applications

Rotational Dynamics is a concept which studies how external forces cause a body to move around its fixed point of axis. The key equation governing this concept is the Newton's second law for linear motion which can be represented as:

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

Where,

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

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

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

Some Common Examples of Rotational motion include:

• Wheel of a Car
• Beyblade
• Ceiling Fan
• Earth's Rotation

To clearly understand the topic of rotational motion, students need to be familiar with sme important terms and there relation with linear motion. Some of these are as follows:

• Torque (τ)
• Moment of Inertia (I)
• Angular Displacement (θ)
• Angular Velocity (ω)
• Angular Acceleration (α)
• Angular Momentum (L)
• Newton’s Second Law of Rotation

Now let's have a detailed look at each of these terms frequently used in this chapter which will strengthen your basic understanding of rotational motion.

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.

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

The torque $\vec{\tau }$ about a point can be defined as:

$\vec{\tau }$= rFsinθ

where,

$\vec{r}$ = position vector from the axis to the point of application of force $\vec{F}$ .

*Note: θ here is the angle of the force applied. For a fixed axis, only the component of torque along the axis contributes to rotational motion.

The moment of inertia $I$ is the rotational equivalent of mass in linear motion. The exact use of moment of inertia is to calculate the resistance shown by the object while changing its motion. Moment of inertia is referred to the ability of the body to oppose the change in it's rotational 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: $I=\int r^{2}dm$ .

Angular Displacement is referred to the angle at which the object rotates around the fixed point of axis. It is a vector quantity and is measured in radians.The formula for angular displacement ia categorized into two types:

1. Arc/Curve length

This is used to calculate the distance travelled along a circular path. Mathematical representation can be given by:

θ = s/r

Where,

θ = angle of displacement

s = length of the curve/arc

r = radius

2. Object's Position

This used to measure the position of the object from initial position to the final position. Mathematical representation can be given by:

θ = θf - θi

Where,

θ = angle of displacement

θi = angle of initial position

θf = angle of final position

Angular Velocity is the rate of change of angular displacement of the object. It can also be defined as the measure of the speed at which a particular object is rotating.

It is a vector quantity and can be represented by the equation:

ω = dθ/dt

Where,

ω = angular velocity

dθ/dt = rate of change

SI unit of angular velocity is radian per seconds.

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

α = dω/dt

Where,

α = angular acceleration

ω = angular velocity

t = time

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