Introduction
Ace Chapter 3, Circular and Rotational Motion (New Course 2026-2027), with HSA Notes super concise, exam-focused notes designed to help students aim for 90 plus marks in their Punjab Board exams.
This page covers angular kinematics, torque, moment of inertia, and their real-world applications, prepared strictly according to the PCTB/PECTAA New Course syllabus, along with fully solved MCQs, short questions, and numericals.
This chapter extends the linear motion concepts from Chapter 2 into circular and rotational systems, introducing angular displacement, velocity, and acceleration, along with the physical quantities unique to rotation, such as torque and moment of inertia. It is one of the more numerical-heavy chapters in the new course, so thorough practice of the solved problems below is essential before attempting past papers.
Why HSA Notes?
- Super concise and exam-focused
- Key definitions highlighted
- All formulas boxed
- Easy step-by-step solved MCQs, shorts, and numericals
- Conceptual clarity points
- Diagrams included
- Quick revision bullets
- Free PDF download
Exercises Solved
This chapter’s exercise solutions from HSA Notes cover every question type set across all Punjab Boards, with full working shown for each answer.
Solved MCQs
1. The ratio of angular speed of the minute hand to the hour hand of a clock is: 12:1
Solved Short Questions
1. Prove that v = rω.
Starting from the relation between arc length and angle, s = rθ, differentiating both sides with respect to time gives:
v = ds/dt = r(dθ/dt) = rω
2. Prove that a = rα.
Starting from v = rω, differentiating both sides with respect to time gives:
a = dv/dt = r(dω/dt) = rα
Solved Numerical Problems
1. Laser Beam Divergence: For a laser beam with spot diameter d = 2.5 m at a distance D = 3.8 × 10⁸ m (Earth to Moon), the angular divergence is:
θ = d/D = 2.5 / 3.8 × 10⁸ = 6.6 × 10⁻⁹ rad
2. Car Wheel Rotation: A car travels at v = 108 km/h = 30 m/s with wheel radius r = 0.3 m. The angular velocity is ω = 100 rad/s, and the frequency of rotation is approximately f ≈ 16 rev/s.
3. Motor Stopping: A motor initially rotating at ω₀ = 1800 rpm = 30 rev/s decelerates to rest over t = 20 s. The total angle turned through is θ = 300 rev.
4. Vertical Circular Motion, Maximum Speed: For an object of mass m = 0.5 kg moving in a vertical circle of radius r = 0.5 m, with maximum tension T_max = 35.6 N, the maximum speed is approximately v ≈ 5.5 m/s.
5. Flywheel Torque: A flywheel with moment of inertia I = 38 kg·m² decelerates from ω₀ = 2100 rpm = 70π rad/s to rest over t = 180 s. The torque required is approximately τ ≈ 44 N·m.
6. Moment of Inertia of a Sphere: For a solid sphere of mass M = 200 kg and radius R = 0.3 m, using I = (2/5)MR², the moment of inertia is I = 7.2 kg·m².
7. Satellite Orbital Speed: For a satellite orbiting at radius r = 6.6 × 10⁶ m, the orbital speed is approximately v ≈ 7.77 km/s.
8. Artificial Gravity: For a rotating space station of radius r = 20 m with angular velocity ω = 0.5 rad/s, the artificial gravity produced is a = 5 m/s².
9. Moment of Inertia from Angular Momentum: Given angular momentum L = 10 kg·m²/s and angular velocity ω = 2 rad/s, the moment of inertia is I = 5 kg·m².
10. Diver’s Rotation: A diver changes moment of inertia from I₁ = 18 kg·m² to I₂ = 3.6 kg·m², rotating at ω₂ = 4π rad/s for t = 1.5 s. The total angle turned through is θ = 0.6 rev.
View the full PDF for complete diagrams, tables, and extra MCQs, shorts, and examples.
Why This Chapter Matters
Circular and Rotational Motion introduces the angular framework that reappears throughout the rest of the Physics syllabus, particularly in topics involving satellites, artificial gravity, and rotating systems. Numericals in this chapter frequently combine linear and angular quantities in the same question, so being able to move fluently between v = rω, a = rα, and torque-based relationships like τ = Iα is essential for scoring well.
Moment of inertia calculations for different shapes are also heavily tested, making the solved examples above especially valuable for exam preparation.
Available Across All Punjab Boards and Top Schools
These notes are built for students across all Punjab Boards, including BISE Lahore, Gujranwala, Multan, Faisalabad, Sargodha, Rawalpindi, Dera Ghazi Khan, Bahawalpur, and Sahiwal, and are used at top schools including Aitchison College, Lahore Grammar School, Beaconhouse, The City School, LACAS, Convent of Jesus & Mary, DPS Lahore, Garrison Academy, Bloomfield Hall, and Sadiq Public School Bahawalpur.
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Frequently Asked Questions
How is v = rω derived?
Starting from the arc length relation s = rθ and differentiating both sides with respect to time gives v = ds/dt = r(dθ/dt) = rω.
What is the difference between angular velocity and angular acceleration?
Angular velocity (ω) is the rate of change of angular displacement, while angular acceleration (α) is the rate of change of angular velocity, related to linear acceleration by a = rα.
How is moment of inertia calculated for different shapes?
Moment of inertia depends on both the mass and its distribution relative to the axis of rotation; for a solid sphere it is I = (2/5)MR², with different formulas applying to other shapes like discs, rings, and rods.
Where can I download the official PCTB textbook for this chapter?
The official Class 11 Physics textbook (New Course 2026-2027) is available directly from PCTB
Conclusion
With Chapter 3, Circular and Rotational Motion, fully covered, Class 11 students across all Punjab Boards now have a clear understanding of angular kinematics, torque, moment of inertia, and their real-world applications, along with fully solved MCQs, short questions, and numericals. Practice these solved exercises alongside past papers, and consider joining our online classes for guided, exam-focused preparation, to build the accuracy and confidence needed for your board exam.
Best of luck for your Class 11 Physics exam!