Kinematics of Rotation: Definition, Equations

Kinematics of Rotation: The motion associated with the sliding motion of an object along one or more of the three dimensions, i.e., \(x,\,y\), and \(z\), is known as translational motion. In this type of motion, all body parts move the same distance in a given amount of time. It is of two types: rectilinear and curvilinear. An example of such a motion is: walking on a straight road.

The motion in which an object spins around an internal axis continuously is known as rotational motion. In this type of motion, all parts of a body perform circular motion about a common axis. An example of such a motion is the rotation of the Earth.

The translational and rotational motions have a lot of things in common. These are often considered to be analogous. We have terms like displacement, velocity, and acceleration to describe translational motion. In contrast, we have terms like angular displacement, angular velocity, and angular acceleration to measure the motion of an object in rotation.
The rotational motion of a body along a fixed axis can be understood as similar to the linear motion of an object in translational motion.

In kinematics, we use equations to determine various factors associated with the translational motion of an object. Similarly, we can define the equations of motion for rotational motion, which will help us calculate quantities like the angle of rotation, angular velocity, etc. Let us read in detail regarding the kinematics of rotation here.

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Rotational Motion: Terms

Let us understand the terms involved in rotational motion: Let us consider the case in which an object performs rotational motion about a given fixed axis, as shown in the above figure. A rigid body can be considered to be composed of several particles. Take a particle \(P\) on the rotating object, as shown in the figure below. The object is rotating about a fixed axis that is passing through \(O\). As the object rotates, the particle \(P\) gets displaced from one point to another.
Angular Displacement: Just as displacement is the shortest distance between an object’s initial and final position in linear motion, in the same way, angular displacement is the shortest angle between an object’s initial and final position in a circular motion. It is measured in radians.
Suppose at a time \(? = 0\), the angular displacement of the particle \(P\) is \(0\), and after moving for a time \(?\), its angular displacement becomes \(?\). In that case, the total angular displacement of the particle \(P\) becomes \(\theta\).
If \(r\) is the radius of the circle along with the particle is moving, and if \(s\) be the distance travelled by the particle, then the angular displacement of the particle can be given as:
\(\theta = \frac{s}{r}\)
Angular velocity: Just as velocity is the rate of change displacement with time similarly, angular velocity is defined as the rate of change of angular velocity with time. Thus, mathematically, angular velocity can be given as:
\(\omega = \frac{{d\theta }}{{dt}}\)
or \(\theta = \overrightarrow \omega t\)
Average angular velocity: \(\overrightarrow \omega = \frac{{\omega – {\omega _0}}}{t}\)
It is measured in radians per second.
The velocity \(v\) of a particle moving along a circle of radius \(r\), the angular velocity can be given as:
\(v = \omega \times r\)
Angular acceleration: Just as acceleration is the rate of change of velocity with time similarly, angular acceleration is defined as the rate of change of angular velocity with time. Thus, mathematically, angular acceleration can be given as:
\(\alpha = \frac{{d\omega }}{{dt}}\)
It is measured in radians per second squared.

Kinematics: Equations of Motion for Translational Motion

An object of mass \(m\) starts moving at time \(t = 0\) with initial speed \(u\). After some time \(t = t\), it achieves a final velocity \(v\). Let \(a\) be the constant acceleration of the object throughout the motion, and it covers a distance \(s\) then these quantities are related to each other as:
1. \( v = u + at\)
2. \(s = ut + \frac{1}{2}a{t^2}\)
3. \({v^2} – {u^2} = 2as\)
Just as the terms in translational motion are analogous to the terms in rotational motion, we can write the equations of motion for a particle in rotation.

Kinematics: Equations of Motion for Rotational Motion

Consider an object that starts rotating about a fixed axis at time \(t = 0\) with an initial angular velocity \({\omega _0}\). After some time, \(t = t\) let its angular velocity be \(\omega\). Let it move with a constant angular acceleration \(\alpha \) throughout the rotation and starting from an angular displacement \(\theta_0\) it attains a final angular displacement \(\theta\). The equations of motion for such an object can be given as:
1. \(\omega = {\omega _0} + \alpha t\)
2. \(\theta = {\theta _0} + {\omega _0}t + \frac{1}{2}\alpha {t^2}\)
3. \({\omega ^2} – {\omega _0}^2 = 2\alpha \left( {\theta – {\theta _0}} \right)\)

Problems Based on Kinematics of Rotation

Q.1. A car wheel is rotating with uniform angular acceleration covers \(60\) revolutions in the first five seconds after the start. If the angular acceleration at the end of five seconds is \(x\,\pi \,{\text{rad/}}{{\text{s}}^2},\) find the value of \(x\).
Ans: The initial angular velocity of the wheel, \({\omega _0} = 0\)
The initial angular displacement of the wheel, \({\theta _0} = 0\)
Applying the equations of motion:
\(\theta = {\omega _0}\frac{1}{2}\alpha {t^2}\)
\(\alpha = \frac{{2\theta }}{{{t^2}}}\)
\(\alpha = \frac{{2\left( {60} \right)\left( {2\pi } \right)}}{{{{\left( 5 \right)}^2}}} = 9.6\pi \,{\text{rad/}}{{\text{s}}^2}\)
We are given, \(\alpha = x\;\pi \,{\text{rad/s}}^2\)
On comparing the above two equations,
\(x = 9.6\,{\text{rad/}}{{\text{s}}^2}\).

Q.2. A fan, starting from rest,  takes ten seconds to attain the maximum speed of \(200\,rpm\). Assume constant acceleration, find the time taken by the fan for reaching half the full speed.
Ans: The maximum angular velocity achieved by fan,
\({\omega _m} = 200\,rpm = 200 \times \frac{{2\pi }}{{60}} = \frac{{20}}{3}\pi \,{\text{rad/s}}\)
Initial angular velocity, \(\omega = 0\)
Using the equation of motion,
\({\omega _m} = {\omega _0} + \alpha t\)
Thus, \(\alpha = ({\omega _m} – {\omega _0})/t\)
\(\alpha = \frac{{\frac{{20}}{3}\pi – 0}}{{10}} = \frac{2}{3}\pi \,{\text{rad/}}{{\text{s}}^2}\)
For angular velocity to be half the maximum, \(\omega = \frac{{{\omega _m}}}{2} = \frac{{10}}{3}\pi \,{\text{rad/s}}\)
Using the equation,
\(\omega = {\omega _0} + \alpha t\)
\(\frac{{10}}{3}\pi – 0 = \frac{2}{3}\pi \; \times t\)
\(t = 5\;{\text{s}}\).

Q.3. The wheel of a racing car catches a speed of \(80\;{\text{rad/s}}\) in \(8\) seconds. Find the angular displacement of the car wheel in the process.
Ans: Let \({\omega _0}\) be the initial angular velocity and \(\omega\) be the final angular velocity of the car’s wheels.
We are given that: \({\omega _o} = 0,\,\omega = 80\;{\text{rad/s}}\) and \(t = 8\;{\text{s}}\).
Using the equation of motion:
\(\omega = {\omega _0} + \alpha t\)
Substituting the values into this equation,
\(80 – 0 = \alpha \left( 8 \right)\)
\(\alpha = 10\;{\text{rad/}}{{\text{s}}^2}\)
Using the third equation of motion,
\({\omega ^2} – {\omega _0}^2 = 2\alpha \theta \)
\(\theta = \frac{{{{\left( {80} \right)}^2} – 0}}{{2\left( {10} \right)}}\)
\(\theta = 320\;{\text{rad/s}}\).

Summary

The translational and rotational motions have a lot of things in common. These are often considered to be analogous. We can define the equations of motion for rotational motion, which will help us calculate quantities like the angle of rotation, angular velocity, etc.
Angular displacement \((\theta)\) is the shortest angle between an object’s initial and final position in a circular motion.
Angular velocity \((\omega)\) is defined as the rate of change of angular velocity with time.
Angular acceleration \(\left( \alpha \right)\) is defined as the rate of change of angular velocity with time.

Equations of motion are:
1. \(\omega = {\omega _0} + \alpha t\)
2. \(\theta = {\theta _0} + {\omega _0}t + \frac{1}{2}\alpha {t^2}\)
3. \({\omega ^2} – {\omega _0}^2 = 2\alpha \left( {\theta – {\theta _0}} \right)\)

Frequently Asked Questions on Kinematics of Rotation

Q.1. Define angular velocity.
Ans: Angular velocity is the rate of change of angular displacement per unit time.

Q.2. Give the equation of rotational kinematics, which relates the angular velocity, acceleration, and time.
Ans: \(\omega = {\omega _0} + \alpha t\)
Where \(\omega\) is the final angular velocity, \({\omega _0}\) is the final angular velocity, \(\alpha\) is the angular acceleration, and \(t\) is time.

Q.3. What is the SI unit of angular displacement?
Ans: The SI unit of angular displacement in radians.

Q.4. How is the velocity of a rotating particle related to its angular velocity?
Ans: The angular velocity of the particle is equal to the cross-product of its linear velocity and radius.

Q.5. If a particle is rotating with uniform angular velocity, what will be the angular acceleration of the particle?
Ans: If the angular velocity of the rotating particle is uniform, its angular acceleration will be zero.

Practice Questions on Kinematics of Rotation here

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