Introduction to Rotational Motion and Angular Kinematics

Basics of Rotational Motion

• Perfect Analogy: Rotational motion mirrors linear motion; many formulas and concepts operate as direct analogs (e.g., linear position becomes angular position).
• Rigid Body Idealization: Extended objects are treated as perfectly rigid with no internal flex during rotation.
• Axis of Rotation: Every rotating object moves about a specific origin point ($O$) or axis.
• Arc Length Formula: The distance ($s$) traced by a point at radius ($r$) moving through angle ($\theta$) is expressed as $s = r \times \theta$.
• Angular Units: Radians are the preferred unit in physics.
  • $1 \text{ radian} = \frac{180}{\pi} \approx 57.3^{\circ}$.
  • To convert degrees to radians: multiply by $\frac{\pi}{180}$.
  • One full rotation: $360^{\circ} = 2\pi \, \text{radians}$.

Angular Kinematics

• Angular Displacement ($\Delta \theta$): The change in angular position, given by $\theta_{final} - \theta_{initial}$.
• Angular Velocity ($\omega$): The rate of change of angular position.
  • Instantaneous velocity: $\omega = \frac{d\theta}{dt}$.
  • Newtonian notation: $\dot{\theta}$.
  • Counterclockwise rotation is defined as the positive direction.
• Angular Acceleration ($\alpha$): The rate of change of angular velocity.
  • Instantaneous acceleration: $\alpha = \frac{d\omega}{dt} = \frac{d^2\theta}{dt^2}$.
  • Newtonian notation: $\dot{\omega}$ or $\ddot{\theta}$.
• Kinematic Equations (Constant Acceleration):
  • $\omega_{f} = \omega_{i} + \alpha \times t$
  • $\theta_{f} = \theta_{i} + \omega_{i} \times t + \frac{1}{2} \alpha \times t^2$
  • $\omega_{f}^2 = \omega_{i}^2 + 2 \alpha \times (\theta_{f} - \theta_{i})$
  • $\theta_{f} = \theta_{i} + \frac{1}{2}(\omega_{i} + \omega_{f}) \times t$

Relating Translational and Angular Quantities

• Tangential Velocity ($v$): The linear speed of a point on a rotating object, calculated as $v = r \times \omega$.
• Tangential Acceleration ($a_t$): Calculated as $a_t = r \times \alpha$.
• Centripetal Acceleration ($a_c$): Directed toward the center of the circle, calculated as $a_c = \frac{v^2}{r} = r \times \omega^2$.
• Total Translational Acceleration ($a$): The magnitude of the combined components:
  • $a = \sqrt{a_t^2 + a_c^2} = r \times \sqrt{\alpha^2 + \omega^4}$.

Rotational Kinetic Energy and Moment of Inertia

• Rotational Kinetic Energy ($K_r$): The energy of a rotating object, defined as $K_r = \frac{1}{2} I \times \omega^2$.
• Moment of Inertia ($I$): The rotational analog for mass, representing how mass is distributed relative to the axis.
  • Discrete Point Masses: $I = \sum m_{i} \times r_{i}^2$.
  • Continuous Extended Objects: $I = \int r^2 \, dm$.
• Density and Integration: For non-discrete objects, $dm$ is defined by density:
  • Line density: $dm = \lambda \, dl$.
  • Surface density: $dm = \sigma \, dA$.
  • Volume density: $dm = \rho \, dV$.
• Calculation Examples:
  • Discrete System: A triangle of masses ($0.3\,kg$, $0.1\,kg$, $0.2\,kg$) connected by rods ($0.3\,m$, $0.4\,m$, $0.5\,m$) yielded $I_1 = 0.057 \, kg \, m^2$ and $K = 0.46 \, J$ at $4.0 \, \text{rad/s}$.
  • Hollow Cylinder (Inner $r_1$, Outer $r_2$): $I = \frac{1}{2} M (r_1^2 + r_2^2)$.
  • Solid Rod (Center of Mass): $I_{cm} = \frac{1}{12} ML^2$.

Parallel Axis Theorem

• Definition: Relates the moment of inertia about the center of mass ($I_{cm}$) to the moment of inertia about a parallel axis a distance ($d$) away.
• Formula: $I = I_{cm} + M \times d^2$.
• Application Example: Finding the moment of inertia for a slender rod of length $L$ rotated about its end:
  • $I_{end} = I_{cm} + M \times (L/2)^2 = \frac{1}{12} ML^2 + \frac{1}{4} ML^2 = \frac{1}{3} ML^2$.

Questions & Discussion

• Exam Material: The speaker concluded the lecture to address audience questions specifically regarding material for the upcoming exam.