Kinematics of Rotational Motion about a Fixed Axis

Rotational motion is divided into two parts – (1) pure rotational motion (rotation about a fixed axis), (2) combined translational and rotational motion (CTRM). In this article, we will study the kinematics of rotational motion about a fixed axis. Are you ready to master this topic for the IIT-JEE exam?

As you know, rotational motion is similar to translational (linear) motion in many aspects. Even the terms we use in rotational motion are analogous to the terms used in translational motion. Before we discuss in detail the kinematics of rotation about a fixed axis, let’s quickly peek through the following definitions.

Rotational kinematics variables

Translational motion involves five primary variables – position, initial velocity, final velocity, acceleration, and time. And each linear variable has a corresponding rotational variable. The parameters of rotational motion are defined as follows.

Angular distance (θ)

It is the angle at which a particle rotates around a specified axis. θ (theta) is measured in radians.

Angular velocity (ω)

It is the rate of change of angular distance (θ) with respect to time. ω is measured in radians per second.

Angular acceleration (α)

It is the rate of change of angular velocity (ω) with respect to time. α is measured in radians per second squared.

Time

Time is time, irrespective of whether the particle is showing translational motion or rotational motion.

Basic rotational motion  equations

 Let’s suppose a particle is undergoing rotational motion about a fixed axis. The particle moves from one point to another, such that the angular displacement is θ.

By definition, angular velocity (ω) is the rate of change of angular displacement. So, mathematically,

ω = dθ/dt

and, similarly,

Angular acceleration α = dω/dt

Since rotational motion parameters are analogous to linear motion variables, rotational motion equations will be similar to kinematical equations for linear motion.

Let’s recall equations for linear motion with constant acceleration.

v = v0+ at

x = x0 + v0t + (1/2) at²

v² = v02+ 2a (x – x0)

where, x0 is the initial displacement, v0 is the initial velocity, x and v are the displacement and velocity of the particle at any given time respectively, a is the acceleration, and t is the time.

The corresponding kinematic equations for rotational motion will be –

ω = ω0+ αt

θ = θ0 + ω0t + (1/2) αt²

ω² = ω02+ 2α (θ – θ0)

Some other rotational motion formulas

Angular displacement θ = x/r

Angular velocity ω  = v/r

Angular acceleration α = a/r

Average angular velocity ωavg = (ω0 + ω)/2

No. of rotations of a particle at any given time N = θ/2π

where x, v, a are linear motion parameters, and r is the radius of curvature of the circular path.

Table showing the analogous relationship between translational and rotational motion

Rotational motion examples

Some common rotational motion examples include rotation of the earth around its axis, blender, drilling machine, the motion of a spinning top, wheels of the motorcycle, merry-go-round, and more.

• The constant angular deceleration of a merry-go-round is 4.0 rad/s2. (a) Find the angle through which the merry-go-round turns as it comes to rest from an angular speed of 440 rad/s. (b) Find the time required for the merry-go-round to halt.

Solution (a):

α = – 4.0 rad/s2

ω0 = 440 rad/s

ω = 0

θ =?

Using this rotational kinematics equation: ω² = ω02+ 2αθ, we get,

0 = (440)2 + 2(- 4)(θ)

=> θ = 24200 rad = 2.42 x 104 rad

Solution (b):

α = – 4.0 rad/s2

ω0 = 440 rad/s

ω = 0

t =?

Using ω = ω0+ αt, we get

0 = 440 + (- 4) t

=> t = 110s

• Many years ago, people listened to music stored on phonograph records. The records turned at a rate of 40 revolutions per minute. Express the rate of rotation in rad/s.

Solution:

We know that

ω = 2πf

=> ω = [2π(40)]/60 rad/s = 4.19 rad/s

Problem-solving strategies for rotational motion kinematics about a fixed axis

Solving questions related to rotational motion kinematics is easy. Here are a few pointers that may be of great help while solving numerical problems.

• Examine whether the numerical involves rotational motion without considering mass and the force that affects the movement.

• Identify the given and unknown variables.

• Identify the appropriate rotational kinematics equation(s) for solving.

• Substitute the given values with the units, and solve the equation to get the required answer.

• Check the calculations, and speculate if your answer is reasonable.

Conclusion

In a nutshell, kinematics is the description of motion. If you know translational motion, the rotational motion will be a piece of cake for you. All you have to do is substitute all the translational motion parameters with rotational motion variables. You can read more articles on translation motion on the Unacademy website.

So far, we have learned about the relationship between various rotational parameters, the analogous relationship between translational and rotational motion, and strategies to solve rotational kinematics problems. Below are a few more questions that may help you with the topic.