# Kinematics of Rotational Motion Around a Fixed Axis

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The rotational motion of the body is analogous to its translational motion. Also, the terms that are used in rotational motion such as the angular velocity and angular acceleration are analogous to the terms velocity and acceleration that are used in translational motion. Thus, we can say that the rotation of a body about a fixed axis is analogous to the linear motion of a body in translational motion. In this section, we will discuss the kinematics kinematic quantities in rotational motion like the angular displacement θ, angular velocity ω angular acceleration α respectively corresponding to kinematic quantities in translational motion like displacement x, velocity v and acceleration a.

## Rotational Kinematics Equations

Let us consider an object undergoing rotational motion about a fixed axis, as shown in the figure, and take a particle P on the rotating object for analyzing its motion. Now as the object rotates about the axis passing through O, the particle P gets displaced from one point to another, such that the angular displacement of the particle is θ.

If at time t = 0, the angular displacement of the particle P is 0 and at time t, its angular displacement is equal to θ, then the total will be θ in time interval t.

Similar to velocity, the rate of change of displacement of the angular velocity is the rate of change of angular displacement with time.

Mathematically, angular velocity,

w = dθ/dt

Further, Similar to acceleration that rate of change velocity the angular acceleration of the particle P is defined as the rate of change of angular velocity of the object wrt time.

Mathematically, angular acceleration,

α = dω/dt

Hence, we see that the kinematic quantities in the rotational motion of the object P are angular displacement(θ), the angular velocity(ω) and the angular acceleration(α) that corresponds to displacement(s), velocity(v) and acceleration(a) in linear or translational motion.

### Kinematic Equations of Rotational Motion

We have already learned in the kinematics equations of linear or translational motion with uniform acceleration.

The three equation of motion was,

v = v0+ at

x = x0 + v0t + (1/2) at²

v² = v02+ 2ax

Where xis the initial displacement and v0 is the initial velocity of the particle.v and x are velocity and displacement respectively at any time t and is the constant acceleration throughout the linear motion. Here initial means t = 0. Now, this equation corresponds to the kinematics equation of the rotational motion as well because we saw above how the kinematics of rotational and translational motion was analogous to each other.

ω = ω0+ αt

θ = θ0 + ω0t + (1/2) αt²

ω² =  ω0² + 2α (θ – θ0)

Where  θis the initial angular displacement of the rotating particle or body, ω0 is the initial angular velocity and α is the constant angular acceleration of the body while ω and θ is the angular velocity and displacement respectively at any time t after the start of motion.

We come across many days today as examples of the relation between the kinematics of rotating body and its translational motion, one of which is if a motorcycle wheel has a large angular acceleration for a fairly long time, it is spinning rapidly and rotates through many revolutions. Thus we can say that, if the angular acceleration of the wheel is large for a long period of time t, then the final angular velocity ω and angle of rotation θ are also very large. The rotational motion of the wheel is analogous to the motorcycle’s large transnational acceleration produces a large final velocity, and also the distance traveled will be large. Also, we can relate the angular displacement θ and translation displacement by equation

S = 2πrN

Where N is the number of a complete rotation of particle chosen at any point on the wheel

N = θ/2π

Q1. How do we Relate Quantities of Rotational Kinematics and Translational Kinematics?

Ans: We can relate the translational kinematic quantities, such as displacement, velocity, and acceleration to rotational motion quantities angular displacement, angular velocity, and angular acceleration respectively as they have a direct analogy with each other.

The three equation of motion were,

v = v0 + at is analog to ω = ω0 + αt

x = x0 + v0t + (1/2) at² is analog to θ = θ0+ ω0t + (1/2) αt²

v² = v02 + 2ax is analog to ω² =  ω0² + 2α (θ – θ0)

Q2. What is Rotational Motion?

Ans: Rotational motion is the motion of a rigid body that takes place in such a way that all of its particles move in circles about a fixed axis with a common angular velocity; and angular acceleration also, the rotation of a particle about a fixed point in space. Together these particles constitute the rotational motion of the rigid body.

Q3. What is the Main Difference Between Circular Motion and Rotational Motion About Fixed Axes?

Ans: The main difference between Circular motion and rotational motion is that the circular motion is just a special case of rotational motion, where the distance between the body's center of mass and the axis of rotation always remains fixed. Rotational motion is based around the idea of rotation of a body about its center of mass. Thus we can say that circular motion is a special type of rotational motion.

Q4. State the Difference Between Translational and Rotational Motion?

Ans: In translational motion, objects move through space i.e. The motion by which a body shifts from one point in space to another, while rotational motion involves objects changing orientation. But there are interesting differences between these motions i.e. in rotational motion with constant tangential velocity is considered as accelerated motion because there the direction of the velocity is changing continuously.

Q5. Give Four Examples of Rotational Motion Around us.

Ans: i) The motion of a spinning top, ii) The spinning of the earth on the axis, iii) Turning off the blades of a fan, iv) Rotation of a windmill or phirki are examples of rotational motion around us.