Rotational motion mirrors linear motion, but with its own unique twists. Instead of force and mass, we deal with torque and moment of inertia. These concepts help us understand how objects spin and rotate in the world around us.

Newton's Second Law for rotation, $\sum \tau = I \alpha$, is the key to solving rotational problems. By analyzing torques and moments of inertia, we can predict how objects will accelerate rotationally, just like we do with linear motion.

Rotational Motion Fundamentals

Newton's Second Law in rotation

• $\sum \tau = I \alpha$ relates net torque, moment of inertia, and angular acceleration
• Mirrors linear motion equation $\sum F = ma$
• Application process involves identifying torques, determining moment of inertia, and calculating angular acceleration
• Rotational kinematics equations describe motion:
  • $\omega = \omega_0 + \alpha t$ (angular velocity)
  • $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$ (angular displacement)
  • $\omega^2 = \omega_0^2 + 2\alpha \theta$ (relates angular velocity and displacement)

Torque and net torque

• Rotational force causing angular acceleration
• Calculated using $\tau = r F \sin \theta$ where r is distance from axis, F is force magnitude, θ is angle between vectors
• Vector quantity with direction determined by right-hand rule
• Net torque found by summing individual torques, considering direction (clockwise or counterclockwise)
• Measured in Newton-meters (N·m)

Moment of Inertia and Rotational Dynamics

Moment of inertia concept

• Measures object's resistance to rotational acceleration
• Analogous to mass in linear motion
• Depends on mass distribution and rotation axis
• Determines rotational ease and affects angular acceleration
• Relates to angular momentum through $L = I \omega$

Moment of inertia calculations

• General formula: $I = \sum m_i r_i^2$
• Common shapes:
  • Solid sphere: $I = \frac{2}{5} M R^2$
  • Thin rod (end pivot): $I = \frac{1}{3} M L^2$
  • Thin rod (center pivot): $I = \frac{1}{12} M L^2$
  • Thin hoop: $I = M R^2$
  • Solid cylinder: $I = \frac{1}{2} M R^2$
• Parallel axis theorem: $I = I_{CM} + M d^2$ for off-center rotations
• Composite objects require summing individual component moments of inertia

Rotational dynamics analysis

• Process: identify external torques, determine total moment of inertia, apply $\sum \tau = I \alpha$, solve for unknowns
• Conservation of angular momentum: $I_1 \omega_1 = I_2 \omega_2$
• Rotational kinetic energy: $K_r = \frac{1}{2} I \omega^2$
• Work-energy theorem for rotation: $W = \Delta K_r$
• Power in rotational motion: $P = \tau \omega$