⚙️AP Physics C: Mechanics Unit 5 – Rotation in AP Physics C: Mechanics

Key Concepts

• Rotation involves objects turning about a fixed axis or point
• Angular displacement ($\theta$) measures the angle through which an object rotates
• Angular velocity ($\omega$) represents the rate of change of angular displacement
• Angular acceleration ($\alpha$) describes the rate of change of angular velocity
• Torque ($\tau$) is the rotational equivalent of force, causing angular acceleration
• Moment of inertia ($I$) is the rotational analog of mass, quantifying an object's resistance to rotational motion
• Rotational kinetic energy ($K_r$) depends on an object's moment of inertia and angular velocity
• Angular momentum ($L$) is conserved in the absence of external torques

Angular Kinematics

• Angular displacement ($\theta$) is measured in radians (rad) or degrees ($^\circ$)
  • One full rotation equals $2\pi$ radians or 360$^\circ$
• Angular velocity ($\omega$) is typically expressed in radians per second (rad/s) or revolutions per minute (rpm)
  • $\omega = \frac{d\theta}{dt}$
• Angular acceleration ($\alpha$) is the derivative of angular velocity with respect to time
  • $\alpha = \frac{d\omega}{dt}$
• Rotational motion equations are analogous to linear motion equations, with $\theta$, $\omega$, and $\alpha$ replacing $x$, $v$, and $a$
  • $\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2$
  • $\omega = \omega_0 + \alpha t$
  • $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$
• Tangential velocity ($v_t$) and acceleration ($a_t$) relate to angular quantities through the radius ($r$)
  • $v_t = r\omega$
  • $a_t = r\alpha$

Torque and Rotational Dynamics

• Torque ($\tau$) is the product of the force ($F$) and the perpendicular distance ($r_\perp$) from the axis of rotation to the line of action of the force
  • $\tau = Fr_\perp = F r \sin\theta$
• Net torque ($\sum \tau$) is the sum of all torques acting on an object
• Newton's Second Law for rotational motion states that the net torque equals the product of the moment of inertia ($I$) and angular acceleration ($\alpha$)
  • $\sum \tau = I\alpha$
• Equilibrium occurs when the net torque is zero, resulting in no angular acceleration
• The torque due to gravity depends on the object's weight ($mg$) and the perpendicular distance from the pivot to the center of mass ($r_\perp$)
  • $\tau_g = mgr_\perp$

Rotational Energy and Momentum

• Rotational kinetic energy ($K_r$) is given by $K_r = \frac{1}{2}I\omega^2$
• Total kinetic energy is the sum of translational and rotational kinetic energies
  • $K_{total} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$
• Work-energy theorem applies to rotational motion, with torque replacing force and angular displacement replacing linear displacement
  • $W = \int \tau d\theta$
• Power in rotational motion is the product of torque and angular velocity
  • $P = \tau\omega$
• Angular momentum ($L$) is the product of the moment of inertia and angular velocity
  • $L = I\omega$
• The rate of change of angular momentum equals the net torque
  • $\frac{dL}{dt} = \sum \tau$

Moment of Inertia

• Moment of inertia ($I$) depends on the object's mass distribution and the axis of rotation
• For a point mass ($m$) at a distance ($r$) from the axis of rotation, $I = mr^2$
• Parallel axis theorem allows calculation of moment of inertia about any parallel axis, given the moment of inertia about the center of mass ($I_{cm}$) and the distance ($d$) between the axes
  • $I = I_{cm} + md^2$
• Radius of gyration ($k$) is the distance from the axis of rotation at which all mass could be concentrated without changing the moment of inertia
  • $I = mk^2$
• Moments of inertia for common shapes (thin rod, thin rectangular plate, solid cylinder, thin-walled hollow cylinder, solid sphere, and thin-walled hollow sphere) are given in terms of mass and dimensions

Angular Momentum Conservation

• Angular momentum is conserved in the absence of external torques
  • $L_i = L_f$ or $I_i\omega_i = I_f\omega_f$
• Conservation of angular momentum explains phenomena such as a spinning figure skater or a cat always landing on its feet
• When a system's moment of inertia changes, its angular velocity must change to conserve angular momentum
  • Pulling arms and legs inward during a spin increases angular velocity
• Precession of a gyroscope or spinning top is a consequence of the conservation of angular momentum
• Angular momentum is a vector quantity, with direction determined by the right-hand rule

Applications and Problem-Solving

• Identify the axis of rotation and the forces causing torques
• Determine the moment of inertia using the parallel axis theorem or the radius of gyration when necessary
• Apply Newton's Second Law for rotational motion to solve for unknown quantities
• Use conservation of angular momentum when no external torques are present
• Combine rotational and translational motion concepts for objects rolling without slipping
  • $v_{cm} = r\omega$
• Solve problems involving rotational work, energy, and power
• Analyze the motion of connected objects, such as gears and pulleys
• Apply angular momentum conservation to collisions and explosions

Common Mistakes and Tips

• Pay attention to the axis of rotation and the perpendicular distance when calculating torques
• Remember that the moment of inertia depends on the axis of rotation and the mass distribution
• Use the right-hand rule consistently to determine the direction of angular quantities
• Be cautious when using the parallel axis theorem; ensure the axes are indeed parallel
• Consider both translational and rotational motion when analyzing rolling objects
• Check the units of angular quantities (radians vs. degrees, radians per second, etc.)
• Practice solving problems with various contexts and complexity to develop problem-solving skills
• Review the relationships between angular and linear quantities, as well as their rotational analogs
• Understand the limitations and assumptions of the models used in rotational motion problems