Last Updated: June 1, 2026

📋 Table of Contents
What is Rotational Motion Revision Notes?
Introduction
1. Centre of Mass (CoM)
2. Torque and Angular Momentum
3. Moment of Inertia (I): Rotational Mass
4. Moment of Inertia Master-Sheet
5. Dynamics of Rolling Motion
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics
📚 Related Topics

📋 Table of Contents

What is Rotational Motion Revision Notes?
Introduction
1. Centre of Mass (CoM)
- Derivation: CoM of a 2-Particle System
2. Torque and Angular Momentum
3. Moment of Inertia (I): Rotational Mass
- Parallel Axes Theorem (Proof)
- Perpendicular Axes Theorem (Proof)
4. Moment of Inertia Master-Sheet
5. Dynamics of Rolling Motion
- Condition for Pure Rolling:
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics

Rotational Motion Class 11 Physics Revision — JEE & NEET 2026 Grandmaster Guide

What is Rotational Motion Revision Notes?

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Torque: τ = r × F = Iα. Turning effect of force.

Moment of Inertia (I): Σ mr². Mass distribution about axis.

Angular Momentum (L): L = r × p = Iω. Conserved if net external torque is zero.

Theorems:

Parallel: I = I_cm + Md²

Perpendicular: I_z = I_x + I_y

Rolling motion: v_cm = ωR. Total K.E. = ½mv_cm² + ½Iω². 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

While the previous chapters dealt with the motion of "point masses," reality involves systems of particles and extended "rigid bodies." In Rotational Motion, every point on a body moves and a circle around an axis, creating a symphony of motion that requires new concepts: Centre of Mass, Torque, n Moment of Inertia. This chapter is widely considered one of the most challenging and class 11 Physics, but it is also the most rewarding for those aiming for top scores and JEE and NEET. In this "Comprehensive" guide, we provide exhaustive derivations for the Parallel and Perpendicular Axes Theorems, the relation between Torque and Angular Momentum, n the physics of pure rolling motion.

1. Centre of Mass (CoM)

The Centre of Mass is the unique point where the entire mass of a system may be considered to be concentrated for describing its translational motion.

Derivation: CoM of a 2-Particle System

Consider two masses m1 n m2 at positions x1 n x2. The position of the CoM is defined as: X_com = (m1 x1 + m2 x2) / (m1 + m2)

For an extended body (continuous mass), use integration: X_com = (1/M) ∫ x dm.

2. Torque and Angular Momentum

I. Torque (τ) - The Rotational Force

Torque is the "turning effect" of a force. Formula: τ = r × F = r F \sinθ. Derivation and Cartesian Coordinates:

Let r = x î + y ĵ n F = Fx î + Fy ĵ.
τ = r × F = (x Fy - y Fx) k̂.

II. Angular Momentum (L)

The rotational equivalent of linear momentum. Formula: L = r × p = I ω.

III. Proof: τ = dL/dt (Rotational Newton's 2nd Law)

L = r × p.
Differentiating wrt time: dL/dt = d(r × p)/dt.
dL/dt = (dr/dt × p) + (r × dp/dt).
Since dr/dt = v n p = mv, the first term (v × mv) = 0 (cross product of parallel vectors).
Since dp/dt = F:
- dL/dt = r × F = τ. (Proven)

3. Moment of Inertia (I): Rotational Mass

Moment of Inertia resists changes and rotational motion. Formula: I = Σ mi ri² = ∫ r² dm.

Parallel Axes Theorem (Proof)

Theorem: I = I_com + Ma² (where I_com is the MoI about an axis through the CoM, n a is the distance between axes).

Proof Logic: By expanding the integral ∫ (x + a)² dm n noting that ∫ x dm = 0 for an axis passing through the CoM, we arrive at the result.

Perpendicular Axes Theorem (Proof)

Theorem: Iz = Ix + Iy (Only for 2D laminar bodies).

Proof Logic: Since r² = x² + y², integrating both sides yields ∫ r² dm = ∫ x² dm + ∫ y² dm, which corresponds to Iz = Iy + Ix.

4. Moment of Inertia Master-Sheet

Body	Axis	MoI (I)
Thin Rod (L)	Centre, Perpendicular	ML² / 12
Circular Ring (R)	Centre, Perpendicular	MR²
Circular Disc (R)	Centre, Perpendicular	MR² / 2
Solid Sphere (R)	Diameter	2/5 MR²
Hollow Sphere (R)	Diameter	2/3 MR²

5. Dynamics of Rolling Motion

When an object rolls, it has both translational and rotational kinetic energy. K_total = K_trans + K_rot = 1/2 Mv² + 1/2 Iω²

Condition for Pure Rolling:

For a body rolling without slipping: v = R ω. If this condition is met, the point of contact is momentarily at rest.

Comprehensive Exam Strategy (Q&A)

Q1: Why are the spokes of a bicycle wheel made thin? Answer: The bulk of the mass is concentrated at the rim (far from the axis). According to I = mr², this maximizes the Moment of Inertia for a given weight, providing the wheel with greater stability and helping it maintain motion once started.

Q2: A ballet dancer pulls her arms and while spinning. Why does her speed increase? Answer: According to the Law of Conservation of Angular Momentum (L = Iω = Constant), when she pulls her arms n, her mass moves closer to the axis, decreasing her Moment of Inertia (I). To keep L constant, her angular velocity (ω) must increase.

Q3: Which will reach the bottom of an incline first: A solid sphere or a circular ring? Answer: The Solid Sphere. Acceleration of a rolling body is a = g \sinθ / (1 + k²/R²). The solid sphere has a smaller radius of gyration (k), leading to a higher acceleration.

Related Revision Notes

Chapter 4: Laws of motion (Inertia Basics)
Chapter 7: gravitation (Orbital Angular Momentum)
Mastering Rotational Mechanics: Advanced Problem Set

Conclusion

Rotational motion is the ultimate test of a physicist's understanding of symmetry and conservation laws. By mastering the mathematical bridge between linear and angular quantities, you gain the ability to analyze everything from a spinning top to the rotation of entire galaxies. Master the parallel axes theorem and the conservation of angular momentum—these are the pillars of advanced mechanics and mechanical engineering. Stay centered, keep your torque high, n maintain your momentum!

Reference: Physics World: The Secrets of Angular Momentum

This post was curated by Jules, Exam Compass Bot, and edited for accuracy y Ayush.

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Continue your revision with these related guides:

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

📚 Related Topics

Continue your revision with these related guides:

🪤 The 5 Mistakes That Cost Marks

Confusing angular velocity and angular frequency: Many students get confused between angular velocity (ω) and angular frequency (ω = 2πf), where f is the frequency of rotation. Remember, angular velocity is the rate of change of angular displacement, while angular frequency is related to the periodic motion.
Forgetting to convert degrees to radians: In rotational motion, angles are often given in degrees, but the formulas require angles in radians. Make sure to convert degrees to radians using the formula: radians = degrees × (π/180).
Incorrectly applying the right-hand rule: The right-hand rule is used to determine the direction of angular quantities like angular velocity and angular momentum. However, students often apply it incorrectly, leading to wrong directions and signs.
Mixing up moment of inertia formulas: There are different formulas for the moment of inertia of various objects, such as rings, disks, and spheres. Students often mix up these formulas or use the wrong one, resulting in incorrect calculations.
Not considering the reference frame: Rotational motion problems often involve different reference frames. Students must ensure they are using the correct reference frame, as the angular velocity and acceleration can vary depending on the chosen frame.

🔁 Last 5 Minutes Box

Kinetic Energy of Rotating Body: K = (1/2)Iω²
Moment of Inertia (I): I = mr² for point mass, I = (1/2)mr² for disc, I = (2/5)mr² for sphere
Torque (τ): τ = r x F = rF\sinθ
Angular Momentum (L): L = Iω = r x p
Equations of Rotational Motion: θ = ω₀t + (1/2)αt², ω = ω₀ + αt, ω² = ω₀² + 2αθ
Relationship between Linear and Angular Quantities: v = rω, a = rα, F = (m/r)τ
Conservation of Angular Momentum: L₁ = L₂, I₁ω₁ = I₂ω₂
Rotational Kinetic Energy: K = (1/2)Iω² = (1/2)(mr²)ω² = (1/2)mv²

Last Updated: June 1, 2026

📋 Table of Contents
What is Rotational Motion Revision Notes?
Introduction
1. Centre of Mass (CoM)
2. Torque and Angular Momentum
3. Moment of Inertia (I): Rotational Mass
4. Moment of Inertia Master-Sheet
5. Dynamics of Rolling Motion
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics
📚 Related Topics

📋 Table of Contents

What is Rotational Motion Revision Notes?
Introduction
1. Centre of Mass (CoM)
- Derivation: CoM of a 2-Particle System
2. Torque and Angular Momentum
3. Moment of Inertia (I): Rotational Mass
- Parallel Axes Theorem (Proof)
- Perpendicular Axes Theorem (Proof)
4. Moment of Inertia Master-Sheet
5. Dynamics of Rolling Motion
- Condition for Pure Rolling:
Comprehensive Exam Strategy (Q&A)
Related Revision Notes
Conclusion
📚 Related Topics

Rotational Motion Class 11 Physics Revision — JEE & NEET 2026 Grandmaster Guide

What is Rotational Motion Revision Notes?

[!TIP] 🚀 2-Minute Quick Recall Summary (Save for Exam Day)

Torque: τ = r × F = Iα. Turning effect of force.

Moment of Inertia (I): Σ mr². Mass distribution about axis.

Angular Momentum (L): L = r × p = Iω. Conserved if net external torque is zero.

Theorems:

Parallel: I = I_cm + Md²

Perpendicular: I_z = I_x + I_y

Rolling motion: v_cm = ωR. Total K.E. = ½mv_cm² + ½Iω². 📥 Download 1-Page Short Notes PDF (Zero-Friction)

Introduction

1. Centre of Mass (CoM)

The Centre of Mass is the unique point where the entire mass of a system may be considered to be concentrated for describing its translational motion.

Derivation: CoM of a 2-Particle System

Consider two masses m1 n m2 at positions x1 n x2. The position of the CoM is defined as: X_com = (m1 x1 + m2 x2) / (m1 + m2)

For an extended body (continuous mass), use integration: X_com = (1/M) ∫ x dm.

2. Torque and Angular Momentum

I. Torque (τ) - The Rotational Force

Torque is the "turning effect" of a force. Formula: τ = r × F = r F \sinθ. Derivation and Cartesian Coordinates:

Let r = x î + y ĵ n F = Fx î + Fy ĵ.
τ = r × F = (x Fy - y Fx) k̂.

II. Angular Momentum (L)

The rotational equivalent of linear momentum. Formula: L = r × p = I ω.

III. Proof: τ = dL/dt (Rotational Newton's 2nd Law)

L = r × p.
Differentiating wrt time: dL/dt = d(r × p)/dt.
dL/dt = (dr/dt × p) + (r × dp/dt).
Since dr/dt = v n p = mv, the first term (v × mv) = 0 (cross product of parallel vectors).
Since dp/dt = F:
- dL/dt = r × F = τ. (Proven)

3. Moment of Inertia (I): Rotational Mass

Moment of Inertia resists changes and rotational motion. Formula: I = Σ mi ri² = ∫ r² dm.

Parallel Axes Theorem (Proof)

Theorem: I = I_com + Ma² (where I_com is the MoI about an axis through the CoM, n a is the distance between axes).

Proof Logic: By expanding the integral ∫ (x + a)² dm n noting that ∫ x dm = 0 for an axis passing through the CoM, we arrive at the result.

Perpendicular Axes Theorem (Proof)

Theorem: Iz = Ix + Iy (Only for 2D laminar bodies).

Proof Logic: Since r² = x² + y², integrating both sides yields ∫ r² dm = ∫ x² dm + ∫ y² dm, which corresponds to Iz = Iy + Ix.

4. Moment of Inertia Master-Sheet

Body	Axis	MoI (I)
Thin Rod (L)	Centre, Perpendicular	ML² / 12
Circular Ring (R)	Centre, Perpendicular	MR²
Circular Disc (R)	Centre, Perpendicular	MR² / 2
Solid Sphere (R)	Diameter	2/5 MR²
Hollow Sphere (R)	Diameter	2/3 MR²

5. Dynamics of Rolling Motion

When an object rolls, it has both translational and rotational kinetic energy. K_total = K_trans + K_rot = 1/2 Mv² + 1/2 Iω²

Condition for Pure Rolling:

For a body rolling without slipping: v = R ω. If this condition is met, the point of contact is momentarily at rest.

Comprehensive Exam Strategy (Q&A)

Related Revision Notes

Chapter 4: Laws of motion (Inertia Basics)
Chapter 7: gravitation (Orbital Angular Momentum)
Mastering Rotational Mechanics: Advanced Problem Set

Conclusion

Reference: Physics World: The Secrets of Angular Momentum

This post was curated by Jules, Exam Compass Bot, and edited for accuracy y Ayush.

📚 Related Topics

Continue your revision with these related guides:

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

🎬 Watch video explanations on YouTube →

📚 Related Topics

Continue your revision with these related guides:

🪤 The 5 Mistakes That Cost Marks

Confusing angular velocity and angular frequency: Many students get confused between angular velocity (ω) and angular frequency (ω = 2πf), where f is the frequency of rotation. Remember, angular velocity is the rate of change of angular displacement, while angular frequency is related to the periodic motion.
Forgetting to convert degrees to radians: In rotational motion, angles are often given in degrees, but the formulas require angles in radians. Make sure to convert degrees to radians using the formula: radians = degrees × (π/180).
Incorrectly applying the right-hand rule: The right-hand rule is used to determine the direction of angular quantities like angular velocity and angular momentum. However, students often apply it incorrectly, leading to wrong directions and signs.
Mixing up moment of inertia formulas: There are different formulas for the moment of inertia of various objects, such as rings, disks, and spheres. Students often mix up these formulas or use the wrong one, resulting in incorrect calculations.
Not considering the reference frame: Rotational motion problems often involve different reference frames. Students must ensure they are using the correct reference frame, as the angular velocity and acceleration can vary depending on the chosen frame.

🔁 Last 5 Minutes Box

Kinetic Energy of Rotating Body: K = (1/2)Iω²
Moment of Inertia (I): I = mr² for point mass, I = (1/2)mr² for disc, I = (2/5)mr² for sphere
Torque (τ): τ = r x F = rF\sinθ
Angular Momentum (L): L = Iω = r x p
Equations of Rotational Motion: θ = ω₀t + (1/2)αt², ω = ω₀ + αt, ω² = ω₀² + 2αθ
Relationship between Linear and Angular Quantities: v = rω, a = rα, F = (m/r)τ
Conservation of Angular Momentum: L₁ = L₂, I₁ω₁ = I₂ω₂
Rotational Kinetic Energy: K = (1/2)Iω² = (1/2)(mr²)ω² = (1/2)mv²