System of Particles and Rotational Motion
📐 Formula & Cheat Sheet (English)
Quick Revision Notes & Formula Sheet
Class 11 Physics | Chapter 7: System of Particles and Rotational Motion
(कणों के निकाय तथा घूर्णी गति)
1. Center of Mass (COM) / द्रव्यमान केंद्र
The Center of Mass of a body or system of particles is defined as a point where the entire mass of the system can be supposed to be concentrated.
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Two-Particle System: $$R_{cm} = \frac{m_1 r_1 + m_2 r_2}{m_1 + m_2}$$
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N-Particle System: $$\vec{R}{cm} = \frac{\sum{i=1}^{n} m_i \vec{r}_i}{M} \quad \text{where } M = \sum m_i$$
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Continuous Mass Distribution: $$\vec{R}_{cm} = \frac{1}{M} \int \vec{r} , dm$$
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Velocity of Center of Mass: $$\vec{v}_{cm} = \frac{m_1 \vec{v}_1 + m_2 \vec{v}_2 + \dots}{M}$$
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Acceleration of Center of Mass: $$\vec{a}_{cm} = \frac{m_1 \vec{a}_1 + m_2 \vec{a}2 + \dots}{M} = \frac{\vec{F}{ext}}{M}$$
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Law of Conservation of Linear Momentum of System: If no external force acts on the system ($\vec{F}{ext} = 0$): $$\vec{a}{cm} = 0 \implies \vec{v}{cm} = \text{Constant}$$ $$\vec{P}{total} = M \vec{v}_{cm} = \text{Constant}$$
2. Vector Product / Cross Product (सदिश गुणनफल)
- Definition: $\vec{A} \times \vec{B} = AB \sin\theta , \hat{n}$ (where $\hat{n}$ is a unit vector perpendicular to the plane containing $\vec{A}$ and $\vec{B}$)
- Properties:
- $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$ (Non-commutative)
- $\hat{i} \times \hat{i} = \hat{j} \times \hat{j} = \hat{k} \times \hat{k} = 0$
- $\hat{i} \times \hat{j} = \hat{k}$, $\hat{j} \times \hat{k} = \hat{i}$, $\hat{k} \times \hat{i} = \hat{j}$
3. Angular Kinematics (कोणीय गतिकी)
| Parameter | Linear Motion | Rotational Motion | Relation |
|---|---|---|---|
| Displacement | $s$ | $\theta$ (radians) | $s = r \theta$ |
| Velocity | $v$ | $\omega = \frac{d\theta}{dt}$ (rad/s) | $v = r \omega$ |
| Acceleration | $a$ | $\alpha = \frac{d\omega}{dt}$ (rad/s²) | $a_t = r \alpha$ |
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Tangential Acceleration: $a_t = r \alpha$
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Centripetal Acceleration: $a_c = \frac{v^2}{r} = \omega^2 r$
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Total Acceleration: $a = \sqrt{a_t^2 + a_c^2}$
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Equations of Rotational Motion (for constant $\alpha$):
- $\omega = \omega_0 + \alpha t$
- $\theta = \omega_0 t + \frac{1}{2} \alpha t^2$
- $\omega^2 = \omega_0^2 + 2\alpha \theta$
4. Torque & Angular Momentum (बल आघूर्ण एवं कोणीय संवेग)
A. Torque ($\vec{\tau}$) [बल आघूर्ण]
- The turning effect of a force about an axis of rotation.
- Formula: $\vec{\tau} = \vec{r} \times \vec{F}$
- Magnitude: $\tau = r F \sin\theta = F \times (\text{Perpendicular distance})$
- SI Unit: $\text{N}\cdot\text{m}$ | Dimensions: $[M^1 L^2 T^{-2}]$
B. Angular Momentum ($\vec{L}$) [कोणीय संवेग]
- The moment of linear momentum about an axis of rotation.
- Formula: $\vec{L} = \vec{r} \times \vec{p}$
- Magnitude: $L = r p \sin\theta = m v r \sin\theta$
- In terms of Moment of Inertia: $L = I \omega$
- SI Unit: $\text{kg}\cdot\text{m}^2/\text{s}$ or $\text{J}\cdot\text{s}$ | Dimensions: $[M^1 L^2 T^{-1}]$
C. Fundamental Relations
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Relation between Torque and Angular Momentum: $$\vec{\tau}_{ext} = \frac{d\vec{L}}{dt}$$
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Law of Conservation of Angular Momentum: If external torque is zero ($\vec{\tau}_{ext} = 0$): $$\vec{L} = \text{Constant} \implies I_1 \omega_1 = I_2 \omega_2$$
5. Moment of Inertia ($I$) & Radius of Gyration ($k$)
A. Moment of Inertia (जड़त्व आघूर्ण)
- Measure of rotational inertia of a body.
- Formula: $I = \sum_{i=1}^{n} m_i r_i^2 = \int r^2 dm$
- SI Unit: $\text{kg}\cdot\text{m}^2$ | Dimensions: $[M^1 L^2 T^0]$
- Factors affecting $I$: Mass of body, distribution of mass, shape & size, position of axis of rotation.
B. Radius of Gyration ($k$) (घूर्णन त्रिज्या)
- Distance from the axis of rotation where the entire mass of the body can be assumed to be concentrated without changing its moment of inertia.
- Formula: $I = M k^2 \implies k = \sqrt{\frac{I}{M}}$
6. Theorems on Moment of Inertia
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Theorem of Parallel Axes (समांतर अक्षों का प्रमेय): $$I = I_{cm} + M d^2$$ (where $I_{cm}$ is moment of inertia about parallel axis passing through COM, $d$ is perpendicular distance between the two axes)
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Theorem of Perpendicular Axes (लंबवत अक्षों का प्रमेय): (Applicable only for 2D planar bodies / laminar sheets) $$I_z = I_x + I_y$$ (where $X, Y, Z$ are mutually perpendicular axes intersecting at a point, $X$ and $Y$ lie in the plane of the object)
7. Moment of Inertia of Standard Uniform Bodies
| Body | Axis of Rotation | Moment of Inertia ($I$) |
|---|---|---|
| Thin Uniform Rod (Length $L$) | Perpendicular to length through Center | $I = \frac{1}{12} M L^2$ |
| Thin Uniform Rod (Length $L$) | Perpendicular to length through One End | $I = \frac{1}{3} M L^2$ |
| Circular Ring (Radius $R$) | Perpendicular to plane through Center | $I = M R^2$ |
| Circular Ring (Radius $R$) | Along Diameter | $I = \frac{1}{2} M R^2$ |
| Circular Disc (Radius $R$) | Perpendicular to plane through Center | $I = \frac{1}{2} M R^2$ |
| Circular Disc (Radius $R$) | Along Diameter | $I = \frac{1}{4} M R^2$ |
| Solid Cylinder (Radius $R$) | Own Axis | $I = \frac{1}{2} M R^2$ |
| Hollow Cylinder (Radius $R$) | Own Axis | $I = M R^2$ |
| Solid Sphere (Radius $R$) | Along Diameter | $I = \frac{2}{5} M R^2$ |
| Spherical Shell / Hollow Sphere | Along Diameter | $I = \frac{2}{3} M R^2$ |
8. Dynamics of Rotational Motion (Rotational Work, Energy & Power)
- Torque: $\tau = I \alpha$
- Rotational Kinetic Energy: $K_{rot} = \frac{1}{2} I \omega^2$
- Work Done by Torque: $W = \int \tau , d\theta = \tau \theta \quad (\text{if } \tau \text{ is constant})$
- Power: $P = \tau \omega$
- Work-Energy Theorem for Rotation: $W = \Delta K_{rot} = \frac{1}{2} I \omega_2^2 - \frac{1}{2} I \omega_1^2$
9. Rolling Motion (लोटनिक गति)
Rolling motion is a combination of pure translation and pure rotation.
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Condition for Pure Rolling (without slipping): $$v_{cm} = R \omega$$
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Total Kinetic Energy in Pure Rolling: $$K_{total} = K_{trans} + K_{rot} = \frac{1}{2} M v_{cm}^2 + \frac{1}{2} I \omega^2$$ $$K_{total} = \frac{1}{2} M v_{cm}^2 \left(1 + \frac{k^2}{R^2}\right)$$
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Acceleration of a Body Rolling Down an Inclined Plane (angle $\theta$): $$a = \frac{g \sin\theta}{1 + \frac{k^2}{R^2}}$$
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Minimum Friction Coefficient for Pure Rolling on Inclined Plane: $$\mu_{min} = \frac{\tan\theta}{1 + \frac{R^2}{k^2}}$$
10. Direct Comparison: Linear vs Rotational Motion
| Feature / Quantity | Linear Motion | Rotational Motion |
|---|---|---|
| Displacement | $s$ | $\theta$ |
| Velocity | $v$ | $\omega$ |
| Acceleration | $a$ | $\alpha$ |
| Mass / Inertia | Mass ($M$) | Moment of Inertia ($I$) |
| Force / Torque | $F = M a$ | $\tau = I \alpha$ |
| Linear / Angular Momentum | $p = M v$ | $L = I \omega$ |
| Work Done | $W = F s$ | $W = \tau \theta$ |
| Kinetic Energy | $K = \frac{1}{2} M v^2$ | $K = \frac{1}{2} I \omega^2$ |
| Power | $P = F v$ | $P = \tau \omega$ |