Exams
Test Series
Previous Year Papers
JEE Main Previous Year Question Paper JEE Advanced Previous Year Papers NEET Previous Year Question Paper CUET Previous Year Papers COMEDK UGET Previous Year Papers UP Polytechnic Previous Year Papers AP POLYCET Previous Year Papers TS POLYCET Previous Year Papers KEAM Previous Year Papers MHT CET Previous Year Papers WB JEE Previous Year Papers GUJCET Previous Year Papers ICAR AIEEA Previous Year Papers CUET PG Previous Year Papers JCECE Previous Year Papers Karnataka PGCET Previous Year Papers NEST Previous Year Papers KCET Previous Year Papers LPUNEST Previous Year Papers AMUEEE Previous Year Papers IISER IAT Previous Year Papers Bihar Diploma DECE-LE Previous Year Papers NPAT Previous Year Papers JMI Entrance Exam Previous Year Papers PGDBA Exam Previous Year Papers AP ECET Previous Year Papers PU CET Previous Year Papers GPAT Previous Year Papers CEED Previous Year Papers AIAPGET Previous Year Papers JKCET Previous Year Papers HPCET Previous Year Papers CG PAT Previous Year Papers SRMJEEE Previous Year Papers BCECE Previous Year Papers AGRICET Previous Year Papers TS PGECET Previous Year Papers MP PAT Previous Year Papers IIT JAM Previous Year Papers CMC Vellore Previous Year Papers ACET Previous Year Papers TS EAMCET Previous Year Papers NATA Previous Year Papers AIIMS MBBS Previous Year Papers BITSAT Previous Year Papers JEXPO Previous Year Papers HITSEEE Previous Year Papers AP EAPCET Previous Year Papers UCEED Previous Year Papers CG PET Previous Year Papers OUAT Previous Year Papers VITEEE Previous Year Papers
Syllabus
JEE Main Syllabus JEE Advanced Syllabus NEET Syllabus CUET Syllabus COMEDK UGET Syllabus UP Polytechnic JEECUP Syllabus AP POLYCET Syllabus TS POLYCET Syllabus KEAM Syllabus MHT CET Syllabus WB JEE Syllabus OJEE Syllabus ICAR AIEEA Syllabus CUET PG Syllabus NID Syllabus JCECE Syllabus Karnataka PGCET Syllabus NEST Syllabus KCET Syllabus UPESEAT EXAM Syllabus LPUNEST Syllabus PUBDET Syllabus AMUEEE Syllabus IISER IAT Syllabus NPAT Syllabus JIPMER Syllabus JMI Entrance Exam Syllabus AAU VET Syllabus PGDBA Exam Syllabus AP ECET Syllabus GCET Syllabus CEPT Syllabus PU CET Syllabus GPAT Syllabus CEED Syllabus AIAPGET Syllabus JKCET Syllabus HPCET Syllabus CG PAT Syllabus BCECE Syllabus AGRICET Syllabus TS PGECET Syllabus BEEE Syllabus MP PAT Syllabus MCAER PG CET Syllabus VITMEE Syllabus IIT JAM Syllabus CMC Vellore Syllabus AIMA UGAT Syllabus AIEED Syllabus ACET Syllabus TS EAMCET Syllabus PGIMER Exam Syllabus NATA Syllabus AFMC Syllabus AIIMS MBBS Syllabus BITSAT Syllabus BVP CET Syllabus JEXPO Syllabus HITSEEE Syllabus AP EAPCET Syllabus GITAM GAT Syllabus UPCATET Syllabus UCEED Syllabus CG PET Syllabus OUAT Syllabus IEMJEE Syllabus VITEEE Syllabus SEED Syllabus MU OET Syllabus
Books
Cut Off
JEE Main Cut Off JEE Advanced Cut Off NEET Cut Off CUET Cut Off COMEDK UGET Cut Off UP Polytechnic JEECUP Cut Off AP POLYCET Cut Off TNEA Cut Off TS POLYCET Cut Off KEAM Cut Off MHT CET Cut Off WB JEE Cut Off ICAR AIEEA Cut Off CUET PG Cut Off NID Cut Off JCECE Cut Off Karnataka PGCET Cut Off NEST Cut Off KCET Cut Off UPESEAT EXAM Cut Off AMUEEE Cut Off IISER IAT Cut Off Bihar Diploma DECE-LE Cut Off JIPMER Cut Off JMI Entrance Exam Cut Off PGDBA Exam Cut Off AP ECET Cut Off GCET Cut Off CEPT Cut Off PU CET Cut Off CEED Cut Off AIAPGET Cut Off JKCET Cut Off HPCET Cut Off CG PAT Cut Off SRMJEEE Cut Off TS PGECET Cut Off BEEE Cut Off MP PAT Cut Off VITMEE Cut Off IIT JAM Cut Off CMC Vellore Cut Off ACET Cut Off TS EAMCET Cut Off PGIMER Exam Cut Off NATA Cut Off AFMC Cut Off AIIMS MBBS Cut Off BITSAT Cut Off BVP CET Cut Off JEXPO Cut Off HITSEEE Cut Off AP EAPCET Cut Off GITAM GAT Cut Off UCEED Cut Off CG PET Cut Off OUAT Cut Off VITEEE Cut Off MU OET Cut Off
Latest Updates
Eligibility
JEE Main Eligibility JEE Advanced Eligibility NEET Eligibility CUET Eligibility COMEDK UGET Eligibility UP Polytechnic JEECUP Eligibility TNEA Eligibility TS POLYCET Eligibility KEAM Eligibility MHT CET Eligibility WB JEE Eligibility OJEE Eligibility ICAR AIEEA Eligibility CUET PG Eligibility NID Eligibility JCECE Eligibility Karnataka PGCET Eligibility NEST Eligibility KCET Eligibility LPUNEST Eligibility PUBDET Eligibility AMUEEE Eligibility IISER IAT Eligibility Bihar Diploma DECE-LE Eligibility NPAT Eligibility JIPMER Eligibility JMI Entrance Exam Eligibility AAU VET Eligibility PGDBA Exam Eligibility AP ECET Eligibility GCET Eligibility CEPT Eligibility PU CET Eligibility GPAT Eligibility CEED Eligibility AIAPGET Eligibility JKCET Eligibility HPCET Eligibility CG PAT Eligibility SRMJEEE Eligibility BCECE Eligibility AGRICET Eligibility TS PGECET Eligibility MP PAT Eligibility MCAER PG CET Eligibility VITMEE Eligibility IIT JAM Eligibility CMC Vellore Eligibility AIMA UGAT Eligibility AIEED Eligibility ACET Eligibility PGIMER Exam Eligibility CENTAC Eligibility NATA Eligibility AFMC Eligibility AIIMS MBBS Eligibility BITSAT Eligibility JEXPO Eligibility HITSEEE Eligibility AP EAPCET Eligibility GITAM GAT Eligibility UPCATET Eligibility UCEED Eligibility CG PET Eligibility OUAT Eligibility IEMJEE Eligibility SEED Eligibility MU OET Eligibility

Dynamics Of Rotational Motion About A Fixed Axis - Testbook

Last Updated on Feb 20, 2025
Download As PDF
IMPORTANT LINKS
System of Particles and Rotational Motion
Relation Between Torque and Moment of Inertia Difference Between Torque and Moment Centrifugal Force Torque Centripetal Force Relation Between Torque and Power Centripetal and Centrifugal Force Difference Between Torque and Power Rotational Kinetic Energy Conservation of Angular Momentum Centre of Mass and Centre of Gravity Rotational Motion Radius of Gyration Angular Momentum Gyroscope Angular Acceleration What is Damping Torque Angular Momentum Parallel and Perpendicular Axis Theorem Rigid Body and Its Dynamics Angular Speed Angular Motion Centre of Gravity Coriolis Force Derivation and Effect Dynamics of Rotational Motion Equilibrium Kinematics of Rotational Motion Relation Between Torque and Speed Translational and Rotational Motion Conservation of Angular Momentum on a Swing Area Moment of Inertia Circular Motion Dynamic Equilibrium How to Calculate Moment of Inertia All Important Centre of Mass Formulas Mass Moment of Inertia Moment of Inertia of a Cone Moment of Inertia of a Circle Moment of Inertia of a Cube Moment of Inertia of a Disc Moment of Inertia of a Rod Moment of Inertia of a Ring Moment of Inertia of a Sphere Moment of Inertia of a Solid Cylinder Moment of Inertia of Ellipse Moment of Inertia of Flywheel Moment of Inertia of Annular Disc Moment of Inertia of Hollow Cone Moment of Inertia of I Section Moment of Inertia of a Square Moment of Inertia of Semicircle Moment of Inertia of Solid Sphere Moment of Inertia of Rectangle Moment of Inertia of T Section Moment of Inertia of a Rectangular Plate Pure Rolling System of Particles
Physical World Units and Measurements Motion in a Straight Line Motion in a Plane Laws of Motion Work Energy and Power Gravitation Mechanical Properties of Solids Mechanical Properties of Fluids Thermal Properties of Matter Thermodynamics Kinetic Theory of Gases Oscillations Waves Electric Charges and Fields Electrostatic Potential and Capacitance Current Electricity Moving Charges and Magnetism Magnetism and Matter Electromagnetic Induction Alternating Current Electromagnetic Waves Ray Optics and Optical Instruments Wave Optics Dual Nature of Radiation and Matter Atoms Nuclei Semiconductor Electronics Earth Science

When studying the motion of rigid bodies, we often come across scenarios where the body is not only translating (moving linearly) but also rotating. The analysis of such motion necessitates a consideration of both linear and angular velocities. To simplify these complex scenarios, we often separate the translational and rotational components of the motion. This article delves into the fascinating world of rotational dynamics, specifically focusing on the motion of objects rotating about a fixed axis.

Table of Contents:

What Exactly is Rotational Motion?

"Rotational motion is the movement of an object along a circular path or in a fixed orbit."

The dynamics of rotational motion mirror those of linear or translational motion. Many of the equations governing rotating objects are akin to those used to describe linear motion. In the context of rotational motion, we only consider rigid bodies, i.e., objects with a definite shape that remains unchanged regardless of the forces acting upon it.


Delving into Rotational Motion About a Fixed Axis

Consider a body rotating about a fixed axis. There is a point on this body with zero velocity, around which the object exhibits rotational motion. This point could be located on the body or somewhere off it. Since the axis of rotation is fixed, we only consider the components of the torques applied to the object that align with this axis as these components induce rotation in the body. The perpendicular component of the torque would attempt to displace the object's axis of rotation.

This displacement would necessitate the emergence of constraining forces, which would counteract the effect of the perpendicular components, thereby maintaining the position of the axis. Since the perpendicular components do not cause any rotation, they are not considered in the calculations. For any rigid body undergoing rotational motion about a fixed axis, we only need to consider the forces that lie in planes perpendicular to the axis.

Why aren't parallel position vectors considered?

Forces parallel to the axis result in torques perpendicular to the axis and are therefore not included in the calculations. Similarly, only the components of the position vector that are perpendicular to the axis are considered. Components of position vectors along the axis lead to torques perpendicular to the axis and are therefore not included in the calculations.

Instances of Rotational Motion

Rotation about a Fixed Point: Some Examples

A few everyday examples of rotation about a fixed point include the spinning of a ceiling fan, the movement of the hands of a clock, and the opening and closing of a door.

Rotation about an Axis of Rotation: Some Examples

When an object rotates about an axis of rotation, it exhibits both translational and rotational motion. A classic example is a ball rolling down an inclined plane. While the ball moves (translates) to the bottom of the plane, it also spins (rotates) about its axis.

Another illustrative example is the Earth's motion. The Earth spins about its axis every day (rotational motion) and also orbits the Sun once every year (translational motion).

Test Series
130.5k Students
NCERT XI-XII Physics Foundation Pack Mock Test
323 TOTAL TESTS | 5 Free Tests
  • 3 Live Test
  • 163 Class XI Chapter Tests
  • 157 Class XII Chapter Tests

Get Started

The Dynamics of Rotational Motion

Moment of Inertia: A Key Concept

The moment of inertia is a measure of an object's resistance to changes in its rotational motion. Denoted by the symbol I, it is expressed in kilogram meter² (kg m 2 .) It can be calculated using the following equation:

I = Mr 2 , where m is the mass of the particle, and r is the distance from the axis of rotation.

The moment of inertia is directly proportional to the mass of the particle; a larger mass results in a higher moment of inertia.

Here's a table showcasing the moment of inertia for various symmetric bodies:

Symmetric body Moment of inertia
Ring with a symmetric axis
Cylinder or disc with a symmetric axis
Uniform sphere
Rod with the axis through the end
Rod with the axis at the centre

Torque: A Fundamental Concept

Torque is the twisting effect resulting from a force applied to a rotating object positioned at a distance r from its axis of rotation. It can be mathematically expressed as:

Angular Momentum: A Crucial Parameter

Angular momentum, denoted by L, quantifies the challenge of bringing a rotating object to a halt. It can be calculated using the following equation:

Comparing Translational and Rotational Motion

The table below presents a comparison between the parameters associated with linear motion and their counterparts in rotational motion:

Serial Number Parameter Linear Motion Rotational Motion
1 Displacement
2 Velocity
3 Acceleration
4 Mass
5 Force
6 Work
7 Kinetic Energy
8 Power
9 Linear Momentum

The Connection Between Rotational Motion and the Work-Energy Principle

The work-energy principle states that the total work done by the sum of all forces acting on an object equals the change in the object's kinetic energy. In the context of rotational motion, this principle is often expressed in terms of torque. According to this interpretation, an object is in equilibrium if the sum of its displacements and rotations equals zero when a force is applied.

Suppose a rigid body experiences a small rotation Δ𝛳. The linear displacement of the body is then Δr = rΔ𝛳, which is perpendicular to r. The work done on the body can therefore be expressed as follows:

ΔW = F perpendicular to Δr

ΔW = F Δr sin 𝜙

ΔW = Fr Δ𝛳 sin 𝜙

ΔW = 𝜏Δ𝛳

If multiple forces are acting on the body, the total work done is:

ΔW = (𝜏1 + 𝜏2 + ……) Δ𝛳

Since Δ𝛳 is the same for all forces, the total work done will be zero:

𝜏1 + 𝜏2 + …… = 0

This demonstrates the validity of the work-energy principle for rotational motion.

The Relationship Between Torque, Moment of Inertia, and Angular Acceleration

Anyone who has ever pushed a merry-go-round or spun a bike wheel intuitively understands the relationship between force, mass, angular velocity, and angular acceleration. As the applied force increases, so does the angular acceleration of the wheel. This relationship can be formalized as follows:

Consider a bike wheel with a force F applied to it, resulting in an angular acceleration 𝛼. Let r be the radius of the wheel. The force is acting perpendicular to the radius, so we can write:

F = ma

where a is acceleration = r𝛼. Therefore, we have

F = mr𝛼

We know that torque is the turning effect of a force, so we can write

𝜏 = Fr

Substituting F = mr𝛼, we get

𝜏 = mr 2 𝛼

This equation is the rotational analogue of F = ma. Here, torque is the analogue of force, angular acceleration is the analogue of acceleration, and rotational inertia (mr 2 ) is the analogue of mass. Rotational inertia is also known as the moment of inertia.

Thus, the relationship between torque, moment of inertia, and angular acceleration can be expressed as:

net 𝜏 = I𝛼

𝛼 = net 𝜏/I

where net 𝜏 is the total torque.

More Articles for Physics

Frequently Asked Questions

Rotational motion can be defined as the motion of an object around a circular path in a fixed orbit.

Ceiling fan rotation, rotation of the minute hand and the hour hand in the clock, and the opening and closing of the door are some of the examples of rotation about a fixed point.

The best example of rotation about an axis of rotation is pushing a ball from an inclined plane. The ball reaches the bottom of the inclined plane through translational motion while the motion of the ball is happening as it is rotating about its axis, which is rotational motion.

The moment of inertia measures the object’s resistance to the change in its rotation.

Torque is the twisting effect of the force applied to a rotating object which is at a position r from its axis of rotation.

No, torque and moment of inertia are not similar. Torque is dependent on the magnitude and direction of the force and on the application point. Whereas the moment of inertia is dependent on the mass and the axis of rotation.

Tangential acceleration, a t is defined as the linear acceleration of a rotating object such that the linear acceleration is perpendicular to the radial acceleration. The SI unit of tangential acceleration is m/s2.

Angular acceleration and tangential acceleration are most of the time considered to be similar, but they are not. Angular acceleration is the change in an object’s angular velocity over time. In contrast, tangential acceleration is defined as the change in the linear velocity of an object over time.

The velocity of an object is constant when the object is moving under translational motion. In contrast, the angular velocity of an object varies when the object is moving under rotational motion. In translational motion mass of an object is considered, whereas in rotational motion moment of inertia of an object is considered.

Report An Error