Rotational motion is used in kinematics to help us to study the motion of things that rotate on a fixed axis, such as the blades of a fan, a spinning wheel, or the Earth itself.
What is Rotational Motion?
Every point on an object moves along a circular route centered on a fixed axis when it rotates around it. Rotational motion is concerned with an object’s ability to rotate, as opposed to linear motion, which is an object moving in a straight line.
To keep things simple, we have to take the best-case situation, in which the axis of rotation remains stationary. This enables us to use angular quantities to describe the motion.
Key Terms in Rotational Motion
1.Angular Displacement (θ): The amount that an object has rotated from its starting location is measured by its angular displacement. Although it is measured in radians (rad) somewhat than meters, it is comparable to linear displacement in linear motion. A wheel’s angular displacement is π rad if it revolves half a circle.
2. Angular Velocity (ω): Angular velocity indicates how quickly an object is rotating. It is defined as the rate of change of angular displacement over time: ω = Δθ / Δt
Angular velocity is measured in radians per second (rad / s).
3. Angular Acceleration (α): Angular acceleration represents the rate at which angular velocity changes over time. It is given by:
α = Δω / Δt Angular acceleration is measured rad/s2
4. Time (t): Similar to linear motion, time is an independent variable that aids in tracking the motion’s evolution in rotating motion.
Equations of Rotational Motion
Rotational motion equations are similar to the equations of linear motion, but they are angular quantities (θ,ω,α) instead of linear ones (x,v,a).
1.First Equation: ω = ω0 + αt
This equation relates angular velocity, initial angular velocity, angular acceleration, and time.
2. Second Equation:
θ = ω0t + 1 / 2αt2
It gives the angular displacement for a given angular acceleration over time.
3. Third Equation: ω2 = ω02 + 2αθ
Without specifically mentioning time, this connects angular velocity, initial angular velocity, angular displacement, and angular acceleration.
In the same way as linear motion equations assume constant acceleration, these equations operate under the assumption of constant angular acceleration.
Relating Linear and Rotational Motion
Rotational motion has close similarity to linear motion. For example:
Linear displacement (x) corresponds to angular displacement (θ).
Linear velocity (v) corresponds to angular velocity (ω).
Linear acceleration (a) corresponds to angular
acceleration (α).
Additionally, for a point on the rotating object located at a distance r from the axis:
The linear velocity is v = r . ω.
The linear acceleration is a = r. α
Applications in Real Life
The kinematics of rotational motion is essential in various fields:
Engineering: Designing gears, turbines, and rotating machinery.
Astronomy: Studying planetary rotations and orbits.
Sports: Analysing the motion of balls, bats, and wheels.
Note :-
How objects spin is provided by the kinematics of rotational motion about a fixed axis. We can effectively study rotational phenomena in the actual world by use of the equations of motion and the ideas of angular displacement, velocity, and acceleration. This issue is also simpler to understand and use because of the close connection with linear motion.
Rotational motion about a fixed axis occurs when an object spins around an axis that does not move. Every point in the object traces a circular path centered on this axis. Examples include a spinning fan or the rotation of the Earth.
The main quantities in rotational motion are:
Angular Displacement (θ): Measures how much an object has rotated, in radians.
Angular Velocity (ω): The rate of change of angular displacement, in radians per second.
- Angular Acceleration (α): The rate of change of angular velocity, in radians per second squared.
For constant angular acceleration, the equations are:
ω = ω0 + αt
θ = ω0t + 1 / 2 αt2
ω2 = ω02 + 2αθ
Here, ω0 is the initial angular velocity, ω is the final angular velocity, α is angular acceleration, θ is angular displacement, and t is time.
Rotational motion has analogous quantities to linear motion:
Angular displacement (θ) corresponds to linear displacement (x).
Angular velocity (ω) corresponds to linear velocity (v).
Angular acceleration (α) corresponds to linear acceleration (a).
For a point at a distance r from the axis:
Linear velocity: v = r.ω
Linear acceleration: a = r.α
Angular displacement is the angle through which an object rotates about its axis, measured in radians. A complete rotation corresponds to 2π radians (360°). It indicates the change in the object’s orientation.
Fans and Wheels: Rotating blades of a fan or the wheel of a vehicle.
Earth’s Rotation: The Earth rotates around its axis, causing day and night.
Gears and Machinery: Rotational motion is used in mechanical devices like engines and turbines.
Sports: A spinning cricket ball or a rolling bicycle tire exhibits rotational motion.
Angular Velocity (ω): Describes how fast an object is rotating and is measured in radians per second (rad/s).
Angular Acceleration (α): Describes how quickly the angular velocity is changing, measured in radians per second squared (rad / s2).