Exam Preparation Notes on Rotational Motion and Dynamics

Rotational Motion

• Non-uniform circular motion:
  • Object moves along a circle, not necessarily at constant speed.

Rotational Kinematics

• Instantaneous Angular Velocity ($w$):
  $w = \frac{d\theta}{dt}$
• Instantaneous Angular Acceleration ($\alpha$):
  $\alpha = \frac{dw}{dt}$
• Definitions:
  • $2\pi \text{ rad} = 360° = 1 \text{ revolution}$
  • Uniform Circular Motion:
  • $w = \text{const.}$ or $\alpha = 0$
  • Uniformly Accelerated Circular Motion:
  • $\alpha = \text{const.}$
• Relations with Translational Quantities:
  • Arc Length ($s$):
  • $s = R\theta$
  • Tangential Velocity ($v_{t}$):
  • $v_{t} = Rw$
  • Tangential Acceleration ($a_{tan}$):
  • $a_{tan} = R\alpha$
  • Centripetal Acceleration ($a_{cp}$):
  • $a_{cp} = \frac{v^2}{R} = Rw^{2}$

Vector Product of Two Vectors

• Definition:
  • Cross Product $\mathbf{a} \times \mathbf{b}$ yields a vector:
  • Magnitude: $\left|\mathbf{a} \times \mathbf{b}\right| = ab \sin(\theta)$
  • Direction: Perpendicular to the plane containing $\mathbf{a}$ and $\mathbf{b}$ (Right-Hand Rule).
• Properties of Cross Product:
  1. Anti-commutative: $\mathbf{a} \times \mathbf{b} = -\mathbf{b} \times \mathbf{a}$
  2. Non-associative: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \neq (\mathbf{a} \times \mathbf{b}) \times \mathbf{c}$
  3. Bilinear:
     • $\alpha \mathbf{a} \times \mathbf{b} + \beta \mathbf{c} \times \mathbf{d} = \alpha (\mathbf{a} \times \mathbf{c}) + \beta (\mathbf{b} \times \mathbf{d})$
• In Cartesian Coordinates:
  • $\mathbf{a} = a{x}\mathbf{i} + a{y}\mathbf{j} + a_{z}\mathbf{k}$
  • $\mathbf{b} = b{x}\mathbf{i} + b{y}\mathbf{j} + b_{z}\mathbf{k}$
  • $\mathbf{a} \times \mathbf{b} = (a{y}b{z} - a{z}b{y})\mathbf{i} + (a{z}b{x} - a{x}b{z})\mathbf{j} + (a{x}b{y} - a{y}b{x})\mathbf{k}$

Returning to Circular Motion

• Angular Velocity Vector ($\mathbf{w}$):
  • Defined as:
  • Magnitude $w$; direction along the rotation axis (Right-Hand Rule).
• Angular Acceleration Vector ($\alpha$):
  • Direction aligned with rotation axis.

Rotational Dynamics

• Torque ($\tau$):
  • Definition: $\tau = R \times F$ where $R$ is the lever arm and $F$ is the force applied.
  • Magnitude: $|\tau| = R F \sin(\theta)$.
  • Units: $\text{N m}$ (Newton-meters).

Moment of Inertia

• Newton's Second Law for Rotational Motion:
  • $\sum \tau = I\alpha$
  • $I = \sum m{i} r{i}^{2}$
  • Moment of Inertia depends on mass distribution relative to the rotational axis.

Determining Moments of Inertia

• Example Calculations:
  1. Two weights on a rod:
  • $I_a = 3 \cdot 2^2 + 6 \cdot 1^2 = 18$
  • $I_b = 3 \cdot 1^2 + 6 \cdot 4^2 = 99$
  1. Thin ring of mass $M$ and radius $R$:
  • $I = MR^2$
  1. Solid sphere of mass $M$:
  • $I = \frac{2}{5}MR^2$.

The Parallel and Perpendicular Axis Theorem

• Parallel Axis Theorem:
  • $I = I_{cm} + Md^{2}$ where $d$ is the distance from the center of mass to the new axis.
• Perpendicular Axis Theorem:
  • For plane objects: $I{z} = I{x} + I_{y}$.

Translational and Rotational Motion

• Motion of any rigid object can be decomposed into:
  • Motion of the center of mass and motion relative to the center of mass.

Example Problems: Rolling and Inclined Planes

1. Rolling Without Slipping:
   • No-slip conditions apply: $a_{cm} = R\alpha$
2. Pulley and Bucket System:
   • Established equations of motion combining rotational dynamics and net torque.

Rotational Kinetic Energy and Work

• Rotational Kinetic Energy Formula:
  • $K_{rot} = \frac{1}{2} I \omega^{2}$
• Work done by torque:
  • $W_{rot} = \tau \theta$ where $\theta$ is the angle in radians.

Conservation of Energy in Rotational Systems

• Total Energy is conserved for systems with both translational and rotational elements.
• Energy equations for systems involving a pulley or incline involve the kinetic energies of the components (translational and rotational).