⚙️AP Physics C: Mechanics (2025) Unit 5 – Torque and Rotational Motion

Key Concepts

• Rotational motion involves objects rotating about a fixed axis rather than moving in a straight line
• Angular displacement ($\theta$) measures the angle through which an object rotates (radians or degrees)
• Angular velocity ($\omega$) represents the rate of change of angular displacement with respect to time (rad/s)
• Angular acceleration ($\alpha$) describes the rate of change of angular velocity with respect to time (rad/s²)
• Torque ($\tau$) is the rotational equivalent of force and causes an object to rotate (N·m)
• Moment of inertia ($I$) quantifies an object's resistance to rotational motion and depends on its mass distribution (kg·m²)
• Angular momentum ($L$) is the rotational analog of linear momentum and is conserved in the absence of external torques (kg·m²/s)

Angular Kinematics

• Angular kinematics describes the motion of rotating objects without considering the forces causing the motion
• Angular displacement ($\theta$) is analogous to linear displacement ($\Delta x$) but measured in radians or degrees
  • One complete revolution corresponds to an angular displacement of $2\pi$ radians or 360 degrees
• Angular velocity ($\omega$) is the rate of change of angular displacement with respect to time ($\omega = \frac{d\theta}{dt}$)
  • Constant angular velocity implies uniform circular motion
• Angular acceleration ($\alpha$) is the rate of change of angular velocity with respect to time ($\alpha = \frac{d\omega}{dt}$)
  • Constant angular acceleration results in uniformly accelerated circular motion
• Kinematic equations for rotational motion are analogous to those for linear motion, with angular quantities replacing linear ones
  • $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
  • $\omega = \omega_0 + \alpha t$
  • $\theta = \theta_0 + \frac{1}{2}(\omega_0 + \omega)t$
  • $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Torque and Rotational Dynamics

• Torque ($\tau$) is the rotational equivalent of force and causes an object to rotate
  • Torque is defined as the cross product of the position vector ($\vec{r}$) and the force vector ($\vec{F}$): $\vec{\tau} = \vec{r} \times \vec{F}$
  • The magnitude of torque is given by $\tau = rF\sin\theta$, where $\theta$ is the angle between $\vec{r}$ and $\vec{F}$
• Net torque ($\sum \tau$) is the sum of all torques acting on an object and determines its rotational acceleration
• Newton's second law for rotational motion states that the net torque on an object equals its moment of inertia times its angular acceleration: $\sum \tau = I\alpha$
  • This is analogous to $\sum F = ma$ for linear motion
• Static equilibrium occurs when both the net force and net torque on an object are zero: $\sum F = 0$ and $\sum \tau = 0$
• Rotational dynamics problems often involve finding the net torque on an object and using it to determine the object's angular acceleration or motion

Rotational Inertia

• Rotational inertia, also known as moment of inertia ($I$), quantifies an object's resistance to rotational motion
  • Objects with larger moments of inertia require more torque to achieve the same angular acceleration
• Moment of inertia depends on the object's mass distribution and the axis of rotation
  • For a point mass: $I = mr^2$, where $m$ is the mass and $r$ is the distance from the axis of rotation
  • For extended objects, moment of inertia is calculated by integrating over the object's mass distribution: $I = \int r^2 dm$
• The parallel axis theorem allows the calculation of an object's moment of inertia about any parallel axis, given its moment of inertia about an axis through its center of mass: $I = I_{CM} + Md^2$
  • $I_{CM}$ is the moment of inertia about the center of mass, $M$ is the total mass, and $d$ is the distance between the parallel axes
• Moments of inertia for common shapes (thin rod, solid cylinder, hollow cylinder, thin rectangular plate) can be derived or looked up in reference tables
• Rotational kinetic energy is given by $K_{rot} = \frac{1}{2}I\omega^2$, analogous to $K = \frac{1}{2}mv^2$ for linear motion

Angular Momentum

• Angular momentum ($\vec{L}$) is the rotational analog of linear momentum and is defined as the cross product of the position vector ($\vec{r}$) and the linear momentum vector ($\vec{p}$): $\vec{L} = \vec{r} \times \vec{p}$
  • For a point mass: $\vec{L} = m\vec{r} \times \vec{v}$
  • For an object rotating about a fixed axis: $L = I\omega$
• The net external torque on a system equals the rate of change of its angular momentum: $\sum \tau_{ext} = \frac{dL}{dt}$
  • This is the rotational equivalent of Newton's second law: $\sum F = \frac{dp}{dt}$
• Angular momentum is conserved in the absence of external torques: if $\sum \tau_{ext} = 0$, then $L = I\omega = \text{constant}$
  • This principle is used to analyze collisions and explosions involving rotating objects
• The angular momentum of a system is the sum of the angular momenta of its individual components: $L_{sys} = \sum L_i$
• Conservation of angular momentum explains phenomena such as the increase in rotational speed when a spinning figure skater pulls their arms in or a diver tucks during a somersault

Energy in Rotational Motion

• Rotational kinetic energy is the kinetic energy associated with an object's rotational motion: $K_{rot} = \frac{1}{2}I\omega^2$
  • 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 done by a torque is the product of the torque and the angular displacement: $W = \tau\theta$
  • This is analogous to $W = Fd$ for linear motion
• The work-energy theorem for rotational motion states that the net work done on an object equals the change in its rotational kinetic energy: $W_{net} = \Delta K_{rot} = \frac{1}{2}I\omega_f^2 - \frac{1}{2}I\omega_i^2$
• Power in rotational motion is the rate at which work is done by a torque: $P = \tau\omega$
  • This is analogous to $P = Fv$ for linear motion
• Conservation of energy can be applied to systems involving both translational and rotational motion
  • For example, a rolling object's total energy (gravitational potential + translational kinetic + rotational kinetic) remains constant in the absence of non-conservative forces

Applications and Real-World Examples

• Rotational motion is ubiquitous in everyday life and engineering applications
• Examples of rotational motion include:
  • Wheels and gears in machines and vehicles
  • Flywheels used for energy storage
  • Turbines in power plants and engines
  • Satellites and planets orbiting in space
• Rotational dynamics is crucial for the design and analysis of:
  • Engines and motors
  • Robotics and automation systems
  • Aerospace and automotive engineering
  • Sports equipment (tennis rackets, golf clubs, bicycles)
• Understanding rotational inertia is important for:
  • Balancing and stability of vehicles and machines
  • Performance optimization in sports (figure skating, diving, gymnastics)
  • Design of flywheels and other energy storage devices
• Conservation of angular momentum explains:
  • The motion of gyroscopes and their use in navigation systems
  • The stability of spinning objects (tops, yo-yos, Frisbees)
  • The formation of spiral galaxies and accretion disks around black holes

Problem-Solving Strategies

• Identify the key concepts and principles relevant to the problem (rotational kinematics, dynamics, conservation laws)
• Draw clear diagrams showing the rotating object, forces, and torques acting on it
  • Use the right-hand rule to determine the direction of torques and angular quantities
• Define a convenient coordinate system and assign positive directions for angular quantities
• Determine the moment of inertia of the object, either by calculation or from reference tables
• Apply the appropriate equations and conservation laws:
  • Rotational kinematics equations for problems involving angular displacement, velocity, and acceleration
  • Newton's second law for rotational motion ($\sum \tau = I\alpha$) for problems involving torques and angular acceleration
  • Conservation of angular momentum ($L = I\omega = \text{constant}$) for problems involving collisions or systems with no external torques
  • Conservation of energy ($\Delta E = 0$) for problems involving gravitational potential energy, rotational kinetic energy, and work done by torques
• Check the units of your answer and ensure they are consistent with the problem's context
• Verify that your solution makes physical sense and aligns with your intuition about the system's behavior