The rotational motion of the body is analogous to its translational motion. Also, the terms that are used in rotational motion such as the angular velocity and angular acceleration are analogous to the terms velocity and acceleration that are used in translational motion. Thus, we can say that the rotation of a body about a fixed axis is analogous to the linear motion of a body in translational motion. In this section, we will discuss the kinematics kinematic quantities in rotational motion like the angular displacement θ, angular velocity ω angular acceleration α respectively corresponding to kinematic quantities in translational motion like displacement x, velocity v and acceleration a.
Rotational Kinematics Equations
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Let us consider an object undergoing rotational motion about a fixed axis, as shown in the figure, and take a particle P on the rotating object for analyzing its motion. Now as the object rotates about the axis passing through O, the particle P gets displaced from one point to another, such that the angular displacement of the particle is θ.
If at time t = 0, the angular displacement of the particle P is 0 and at time t, its angular displacement is equal to θ, then the total will be θ in time interval t.
Similar to velocity, the rate of change of displacement of the angular velocity is the rate of change of angular displacement with time.
Mathematically, angular velocity,
w = dθ/dt
Further, Similar to acceleration that rate of change velocity the angular acceleration of the particle P is defined as the rate of change of angular velocity of the object wrt time.
Mathematically, angular acceleration,
α = dω/dt
Hence, we see that the kinematic quantities in the rotational motion of the object P are angular displacement(θ), the angular velocity(ω) and the angular acceleration(α) that corresponds to displacement(s), velocity(v) and acceleration(a) in linear or translational motion.
Kinematic Equations of Rotational Motion
We have already learned in the kinematics equations of linear or translational motion with uniform acceleration.
The three equation of motion was,
v = v0+ at
x = x0 + v0t + (1/2) at²
v² = v02+ 2ax
Where x0 is the initial displacement and v0 is the initial velocity of the particle.v and x are velocity and displacement respectively at any time t and is the constant acceleration throughout the linear motion. Here initial means t = 0. Now, this equation corresponds to the kinematics equation of the rotational motion as well because we saw above how the kinematics of rotational and translational motion was analogous to each other.
ω = ω0+ αt
θ = θ0 + ω0t + (1/2) αt²
ω² = ω0² + 2α (θ – θ0)
Where θ0 is the initial angular displacement of the rotating particle or body, ω0 is the initial angular velocity and α is the constant angular acceleration of the body while ω and θ is the angular velocity and displacement respectively at any time t after the start of motion.
We come across many days today as examples of the relation between the kinematics of rotating body and its translational motion, one of which is if a motorcycle wheel has a large angular acceleration for a fairly long time, it is spinning rapidly and rotates through many revolutions. Thus we can say that, if the angular acceleration of the wheel is large for a long period of time t, then the final angular velocity ω and angle of rotation θ are also very large. The rotational motion of the wheel is analogous to the motorcycle’s large transnational acceleration produces a large final velocity, and also the distance traveled will be large. Also, we can relate the angular displacement θ and translation displacement by equation
S = 2πrN
Where N is the number of a complete rotation of particle chosen at any point on the wheel
N = θ/2π
FAQs on Kinematics of Rotational Motion Around a Fixed Axis
1. What is meant by the kinematics of rotational motion around a fixed axis?
The kinematics of rotational motion describes the motion of a rigid body rotating about a fixed, stationary axis, without considering the forces or torques that cause the motion. It focuses on analysing three primary variables: angular position (θ), angular velocity (ω), and angular acceleration (α).
2. What are the key variables used to describe the rotation of an object?
Three key variables are used to describe the rotation of an object:
- Angular Position (θ): The angle (in radians) through which a point on the body has turned relative to a reference line.
- Angular Velocity (ω): The rate at which the angular position changes with time (ω = dθ/dt). It tells us how fast the object is spinning.
- Angular Acceleration (α): The rate at which the angular velocity changes with time (α = dω/dt). It tells us if the object's spin is speeding up or slowing down.
3. What are the three fundamental equations of rotational kinematics for constant angular acceleration, as per the CBSE 2025-26 syllabus?
For a body rotating with constant angular acceleration, the motion is described by three equations that are analogous to linear motion:
- ω = ω₀ + αt
- θ = θ₀ + ω₀t + ½αt²
- ω² = ω₀² + 2α(θ – θ₀)
4. What are some common real-world examples of rotational motion around a fixed axis?
Rotational motion around a fixed axis is very common. Key examples include:
- The blades of a ceiling fan rotating around its central motor.
- A potter's wheel spinning on its base.
- The Earth spinning on its axis, which causes day and night.
- A merry-go-round or a Ferris wheel rotating about a central pivot.
- A spinning top rotating about its vertical axis of symmetry.
5. What is the crucial difference between rotational motion and circular motion?
The key distinction lies in what is being described. Rotational motion refers to the movement of an entire extended body turning about an axis. For instance, a spinning turntable exhibits rotational motion. In contrast, circular motion describes the path of a single particle or point. A specific dot on the edge of that spinning turntable undergoes circular motion. Therefore, circular motion is a result of the rotational motion of the body it is part of.
6. How does the analogy between linear and rotational kinematics help in understanding physics?
The analogy is powerful because it allows us to apply a familiar set of rules to a new context. If you understand the concepts of linear displacement (s), velocity (v), and acceleration (a), you can directly map them to angular displacement (θ), angular velocity (ω), and angular acceleration (α). This means the problem-solving structure and the form of the kinematic equations are identical, which significantly simplifies learning and applying the concepts of rotational motion.
7. If a fan rotates at a constant speed, does any part of it have acceleration?
Yes. Even though the fan has a constant angular velocity (ω), its angular acceleration (α) is zero. However, every particle on the blades (except those on the axis) is constantly changing its direction of movement. This change in the direction of linear velocity results in a centripetal acceleration (a꜀ = rω²). This is a type of linear acceleration that points towards the center of rotation, so acceleration is present even at a constant rotational speed.
8. How does a particle's distance from the axis of rotation affect its linear speed?
A particle's linear speed (v) is directly proportional to its perpendicular distance (r) from the axis of rotation. The relationship is given by the formula v = rω, where ω is the angular velocity. This means that for a rigid body like a spinning disc, all points have the same angular velocity, but a point on the outer edge travels a much greater linear distance in the same time and thus has a higher linear speed than a point near the center.
9. What is the physical importance of the 'axis of rotation'?
The axis of rotation is the line that remains fixed while a rigid body rotates. Its importance is that it acts as the reference for all rotational motion. All particles within the body move in circular paths with their centers on this axis. The axis defines the point of zero velocity in the rotating frame, and all rotational quantities like angular velocity and torque are defined with respect to this line.
10. Can an object have rotational motion even if the axis of rotation is outside the object's body?
Yes, this is possible and quite common. A prime example is the Moon rotating about the Earth. The axis of this rotation is a point (the Earth-Moon barycenter) that lies outside the Moon's physical body. Another simple example is a stone being whirled on the end of a string; the stone rotates around your hand, an axis that is not part of the stone itself.
