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.

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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.

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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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