System of Particles and Rotational Motion

• Center of Mass (COM)

  • Definition: Point where entire mass of the system is assumed to be concentrated
  • Two-Particle System: $x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$
  • N-Particle System: $\vec{r}_{cm} = \frac{\sum m_i \vec{r}_i}{M}$
  • Continuous Mass Distribution: $\vec{r}_{cm} = \frac{1}{M} \int \vec{r} , dm$
  • Motion of COM: $M\vec{a}{cm} = \vec{F}{ext}$
  • Conservation of Linear Momentum: If $\vec{F}{ext} = 0$, then $\vec{v}{cm} = \text{constant}$

• Vector Product (Cross Product)

  • Definition: $\vec{A} \times \vec{B} = AB \sin\theta , \hat{n}$
  • Properties:
    • Non-commutative: $\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})$
    • Distributive: $\vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C}$
  • Direction: Determined by Right-Hand Thumb Rule

• Angular Kinematics

  • Angular Displacement ($\theta$): Unit in radians
  • Angular Velocity ($\omega$): $\omega = \frac{d\theta}{dt}$
  • Angular Acceleration ($\alpha$): $\alpha = \frac{d\omega}{dt}$
  • Linear to Angular Relations:
    • Position: $s = r\theta$
    • Velocity: $v = r\omega$
    • Tangential Acceleration: $a_t = r\alpha$
  • Rotational Equations of Motion:
    • $\omega = \omega_0 + \alpha t$
    • $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$
    • $\omega^2 = \omega_0^2 + 2\alpha\theta$

• Torque and Angular Momentum

  • Torque ($\vec{\tau}$):
    • Concept: Turning or rotational effect of force
    • Formula: $\vec{\tau} = \vec{r} \times \vec{F} = r F \sin\theta , \hat{n}$
    • Unit: $\text{N}\cdot\text{m}$
  • Angular Momentum ($\vec{L}$):
    • Concept: Rotational analogue of linear momentum
    • Formula: $\vec{L} = \vec{r} \times \vec{p} = I\vec{\omega}$
    • Unit: $\text{kg}\cdot\text{m}^2/\text{s}$
  • Relation: $\vec{\tau} = \frac{d\vec{L}}{dt}$
  • Law of Conservation of Angular Momentum:
    • If $\vec{\tau}_{ext} = 0$, then $\vec{L} = \text{constant}$
    • Equation: $I_1 \omega_1 = I_2 \omega_2$

• Equilibrium of Rigid Bodies

  • Translational Equilibrium: $\sum \vec{F}_{ext} = 0$
  • Rotational Equilibrium: $\sum \vec{\tau}_{ext} = 0$
  • Principle of Moments: $\text{Load} \times \text{Load Arm} = \text{Effort} \times \text{Effort Arm}$
  • Center of Gravity: Point where total gravitational torque on body is zero

• Moment of Inertia ($I$)

  • Concept: Resistance of a body to rotational change (rotational inertia)
  • Formula: $I = \sum m_i r_i^2$
  • Unit: $\text{kg}\cdot\text{m}^2$
  • Radius of Gyration ($k$): $I = Mk^2 \Rightarrow k = \sqrt{\frac{I}{M}}$
  • Theorems of Moment of Inertia:
    • Theorem of Parallel Axes: $I = I_{cm} + M d^2$
    • Theorem of Perpendicular Axes: $I_z = I_x + I_y$ (for planar objects)
  • Standard Values ($I$):
    • Thin Circular Ring: $MR^2$ (center axis)
    • Circular Disc: $\frac{1}{2}MR^2$ (center axis)
    • Thin Uniform Rod: $\frac{1}{12}ML^2$ (perpendicular through center)
    • Solid Cylinder: $\frac{1}{2}MR^2$ (geometrical axis)
    • Solid Sphere: $\frac{2}{5}MR^2$ (diameter axis)

• Rotational Dynamics & Work-Energy

  • Rotational Newton's 2nd Law: $\vec{\tau} = I\vec{\alpha}$
  • Rotational Kinetic Energy: $K_{rot} = \frac{1}{2}I\omega^2$
  • Work Done: $W = \int \tau , d\theta$
  • Power: $P = \tau \omega$

• Rolling Motion

  • Concept: Combination of Translational and Rotational motion
  • Pure Rolling Condition: $v_{cm} = R\omega$
  • Total Kinetic Energy: $K_{total} = K_{trans} + K_{rot} = \frac{1}{2}M v_{cm}^2 \left(1 + \frac{k^2}{R^2}\right)$
  • Acceleration on Inclined Plane: $a = \frac{g \sin\theta}{1 + \frac{k^2}{R^2}}$