System of Particles and Rotational Motion
1. Center of Mass
- The point where the whole mass of a system is considered to be concentrated.
- For a two-particle system:
\( \vec{R} = \frac{m_1 \vec{r}_1 + m_2 \vec{r}_2}{m_1 + m_2} \) - General formula:
\( \vec{R} = \frac{\sum m_i \vec{r}_i}{\sum m_i} \)
2. Motion of Center of Mass
- Moves as if all external forces act on it.
- Velocity of center of mass:
\( \vec{V}_{cm} = \frac{d\vec{R}}{dt} \)
3. Linear Momentum of a System
- Total linear momentum is the product of total mass and velocity of center of mass.
- \( \vec{P} = M \vec{V}_{cm} \)
4. Vector Product (Cross Product)
- \( \vec{A} \times \vec{B} = AB \sin\theta \hat{n} \)
- Used in torque and angular momentum calculations.
5. Torque and Angular Momentum
- Torque: \( \vec{\tau} = \vec{r} \times \vec{F} \)
- Angular Momentum: \( \vec{L} = \vec{r} \times \vec{p} \)
- \( \frac{d\vec{L}}{dt} = \vec{\tau} \)
6. Equilibrium
- Translational equilibrium: Net force = 0
- Rotational equilibrium: Net torque = 0
7. Moment of Inertia
- Resistance to rotational motion.
- \( I = \sum m_i r_i^2 \)
- Depends on mass and distribution about the axis.
8. Radius of Gyration
- \( K = \sqrt{\frac{I}{M}} \)
- Distance from axis at which whole mass can be assumed to be concentrated.
9. Conservation of Angular Momentum
- If net external torque = 0, angular momentum is conserved.
- \( \vec{L}_\text{initial} = \vec{L}_\text{final} \)