# Kinematics of Rotational Motion

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7.2Kinematics of Rotational Motion

In these revision notes for Kinematics of Rotational Motion, we cover the following key points:

• The meaning of some quantities used as a background in the study of rotation, such as radius of curvature, period and frequency of rotation.
• Quantities involved in kinematics of uniform rotational motion, such as angular displacement and angular velocity.
• Quantities involved in kinematics of non-uniform rotational motion, such as angular acceleration and change in angular velocity.
• The relationship between linear and rotational quantities
• The meaning and formula of centripetal acceleration.
• Equations of all abovementioned quantities (there are more than one equation for each quantity)

## Kinematics of Rotational Motion Revision Notes

A circular (rotational) motion involves the rotation of a particle about a fixed point in space by always keeping the same distance from this fixed point. Therefore, a rotational motion implies moving around a circle whose centre and radius are fixed.

The time neccessary to make one complete revolution around a fixed point is called Period, T. It is measured by the unit of time, i.e. second, [s].

When an object rotates very fast around a fixed point, period results in a very small number. Therefore, to avoid the use of decimals, in such cases it is more suitable using the inverse of period, known as Frequency, f, to represent the time-related phenomena. Frequency is measured in revolutions per second, but this unit is widely recognized as Hertz [Hz] instead. Thus, we have

f = 1/T

and

[1 Hz] = [1 1/s ] = [1 s-1 ]

The simplest case of rotational motion is the uniform circular motion which represents objects moving at the same speed (not velocoty) around a fixed point by maintaining a constant distance from it. The reason why velocity is not constant although its magnitude doesn't change, is because the object's moving direction changes continously when it follows a circular path.

Angle of rotation, φ, is a very important parameter of rotational motion. This angle is measured in radians (rad) and it is also known as "angular displacement". When an object makes N rotations around a fixed point, the angle of rotation is

φ = N × 2π rad

The arc length L represents the distance traveled by the object during its movement when the total angle is φ rad. Its relationship with the angular displacement is given by the formula

L = r × φ

If we divide the angle of rotation by the time this process takes, we obtain another important quantity of rotational motion. It is known as the angular velocity, ω is measured in radians per second, [rad/s]. Its formula is

ω = φ/t

The table below gives the relationship between quantities in kinematics of both linear and rotational uniform motion.

If the rotational motion is uniformly accelerated or decelerated, there is another quantity added. It is known as angular acceleration, α and is measured in [rad/s2].

Equations of uniformly accelerated or decelerated rotational motion and their relationship with those of uniformly accelerated or decelerated linear motion are shown in the table below.

In uniform rotational motion, the velocity is not the same everywhere despite its constant magnitude. This is because the velocity vector changes direction during the rotation. Therefore, we obtain a non-zero difference of velocity vectors which when divided by time gives a non-zero acceleration. Its vector always points towards the centre of curvature. That's why it is called centripetal acceleration, aC. Its formula is:

aC = v2 - v1/∆t

We calculate the magnitude of centripetal acceleration by the formula

aC = v2/r

The unit of centripetal acceleration is [m/s2] although it is an acceleration that occurs only in rotational motion. Therefore, centripetal acceleration acts as a bridge between linear and rotational quantities.

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