In Class 11 Physics, "System of Particles and Rotational Motion" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.
Why do figure skaters spin with breathtaking speed simply by pulling their outstretched arms close to their chest, or why is a wrench with a long handle capable of loosening a frozen bolt that won't budge with bare hands? The Conservation of Angular Momentum and Torque govern all rotational motion.
यह अध्याय क्यों महत्वपूर्ण है
In Class 11 Physics, "System of Particles and Rotational Motion" provides an authoritative, curriculum-verified master resource aligned with the 2026–27 NCERT syllabus.
अध्ययन से पूर्व (आवश्यक ज्ञान)
Vectors and cross products.
Newton's laws of motion from Chapter 4.
Work and kinetic energy from Chapter 5.
इस अध्याय के लक्ष्य
Define Center of Mass (COM) for a two-particle and $N$-particle system: $\vec{R}_{cm} = \frac{\sum m_i \vec{r}_i}{\sum m_i}$.
State and apply the Law of Conservation of Angular Momentum: $\frac{d\vec{L}}{dt} = \vec{\tau}_{ext} = 0 \implies I_1 \omega_1 = I_2 \omega_2$.
Define Moment of Inertia ($I = \sum m_i r_i^2$) and Radius of Gyration ($k = \sqrt{I/M}$).
Calculate rotational kinetic energy ($K_{rot} = \frac{1}{2}I\omega^2$) and total kinetic energy of rolling without slipping: $K = \frac{1}{2}mv_{cm}^2(1 + \frac{k^2}{R^2})$.
अध्याय रूपरेखा एवं प्रगति
11. Center of Mass & System Dynamics
22. Torque & Angular Momentum Conser...
33. Moment of Inertia & Rolling Moti...
सम्पूर्ण सैद्धांतिक एवं वैचारिक अध्ययन
1. Center of Mass & System Dynamics
The Center of Mass (COM) is a unique point where the entire mass of a system may be considered to be concentrated for translational motion. For $N$ particles: $$\mathbf{\vec{R}_{cm} = \frac{\sum_{i=1}^N m_i \vec{r}_i}{\sum m_i} = \frac{1}{M}\sum m_i \vec{r}_i}$$ The center of mass of a system moves as if all external forces were applied directly at that point (internal mutual forces cancel out by Newton's Third Law!).
2. Torque & Angular Momentum Conservation
Rotational analogues of linear dynamics: • Torque: $\mathbf{\vec{\tau} = \vec{r} \times \vec{F} = r F \sin\theta\,\hat{n}}$. (Rotational effort). • Angular Momentum: $\mathbf{\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}}$. • Conservation of Angular Momentum: When net external torque is zero ($\vec{\tau}_{ext} = 0$): $$\mathbf{I_1 \omega_1 = I_2 \omega_2 = \text{constant}}$$ When an ice-skater folds her arms, her moment of inertia $I$ decreases, forcing angular velocity $\omega$ to dramatically increase!
3. Moment of Inertia & Rolling Motion
Moment of Inertia ($I$): The rotational inertia resisting angular acceleration: $\mathbf{I = \sum m_i r_i^2 = M k^2}$ ($k$ is radius of gyration). • Thin ring: $I = MR^2$. • Uniform disc: $I = \frac{1}{2}MR^2$. • Solid sphere: $I = \frac{2}{5}MR^2$. In rolling without slipping ($v_{cm} = R\omega$), total kinetic energy is: $\mathbf{K = \frac{1}{2}M v_{cm}^2 \left(1 + \frac{k^2}{R^2}\right)}$.
System of Particles and Rotational Motion - Key Conceptual & Analytical Model
अध्याय का सार संक्षेप एवं 10 मुख्य निष्कर्ष
मुख्य बिंदु 1
Center of Mass: Point tracking systemic linear momentum under external forces.
Moment of Inertia: Mass distribution metric governing rotational acceleration resistance.
मुख्य बिंदु 5
Rolling Kinetic Energy: Sum of linear translational and angular rotational kinetic energies.
स्व-मूल्यांकन अभ्यास (Check Your Understanding)
मूल वैचारिक स्पष्टता की जांच के लिए नैदानिक प्रश्न। पहले स्वयं हल करें, फिर उत्तर देखें।
1
State the law of conservation of angular momentum. Give two practical examples.
उत्तर एवं व्याख्या देखें
उत्तर: If the net external torque acting on a system is zero, the total angular momentum of the system remains constant ($I\omega = \text{constant}$). Examples: (1) A spinning ballet dancer folds her arms to decrease $I$ and increase spin speed $\omega$, (2) A diver curls into a tuck position during a high dive to execute multiple rapid somersaults. Zero external torque conserves Iω; ballet dancer and diver.
2
Find the torque of a force $\vec{F} = 2\hat{i} - 3\hat{j} + 4\hat{k}\text{ N}$ acting at point $\vec{r} = 3\hat{i} + 2\hat{j} + 3\hat{k}\text{ m}$ about the origin.
What is the radius of gyration of a solid sphere of radius $R$ about its diameter?
उत्तर एवं व्याख्या देखें
उत्तर: For a solid sphere, $I = \frac{2}{5}MR^2$. Set $I = Mk^2 \implies Mk^2 = \frac{2}{5}MR^2 \implies k = \sqrt{\frac{2}{5}}R \approx 0.632R$. k = √(2/5) R.
4
Why is a flywheel with a heavy rim used in automobile engines?
उत्तर एवं व्याख्या देखें
उत्तर: Because concentrating mass at the rim maximizes the moment of inertia ($I = MR^2$). A large moment of inertia resists abrupt rotational speed changes, smoothing out engine power jerks between piston power strokes. Concentrates mass at rim for maximum rotational inertia.
5
A solid cylinder and a solid sphere of equal mass and radius roll down an inclined plane from rest. Which one reaches the bottom first?
उत्तर एवं व्याख्या देखें
उत्तर: Acceleration in rolling is $a = \frac{g\sin\theta}{1 + k^2/R^2}$. For sphere: $k^2/R^2 = 2/5 = 0.4 \implies a = g\sin\theta / 1.4$. For cylinder: $k^2/R^2 = 1/2 = 0.5 \implies a = g\sin\theta / 1.5$. The sphere has greater acceleration and reaches the bottom first! Solid sphere reaches first (smaller rotational inertia).
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