ap physics 1 unit 6 study guides

Key Concepts

• Understand the differences between linear and rotational motion involves objects rotating about a fixed axis
• Familiarize yourself with angular displacement ($\theta$), angular velocity ($\omega$), and angular acceleration ($\alpha$)
• Learn how torque ($\tau$) causes angular acceleration, similar to how force causes linear acceleration
• Recognize that rotational kinetic energy ($K_r$) depends on an object's moment of inertia ($I$) and angular velocity
• Understand that angular momentum ($L$) is conserved in the absence of external torques
• Apply conservation of energy to systems involving both translational and rotational motion (rolling without slipping)
• Solve problems involving rotational dynamics in real-world scenarios (wheels, gears, pulleys)

Angular Motion Basics

• Angular displacement ($\theta$) measures the angle through which an object rotates, typically in radians
  • One full rotation equals $2\pi$ radians
• Angular velocity ($\omega$) is the rate of change of angular displacement with respect to time
  • Measured in radians per second (rad/s)
• Angular acceleration ($\alpha$) is the rate of change of angular velocity with respect to time
  • Measured in radians per second squared (rad/s²)
• Tangential velocity ($v_t$) is the linear velocity of a point on a rotating object perpendicular to the radius
  • Related to angular velocity by $v_t = r\omega$, where $r$ is the distance from the axis of rotation
• Tangential acceleration ($a_t$) is the linear acceleration of a point on a rotating object perpendicular to the radius
  • Related to angular acceleration by $a_t = r\alpha$

Rotational Kinematics

• Rotational kinematics describes the motion of a rotating object using angular displacement, velocity, and acceleration
• Angular displacement can be calculated using $\theta = \theta_0 + \omega t + \frac{1}{2}\alpha t^2$
  • $\theta_0$ is the initial angular displacement
• Angular velocity can be calculated using $\omega = \omega_0 + \alpha t$
  • $\omega_0$ is the initial angular velocity
• These equations are analogous to the linear kinematics equations ($x = x_0 + vt + \frac{1}{2}at^2$ and $v = v_0 + at$)
• Rotational motion can be combined with translational motion (rolling without slipping)
  • The linear velocity at the bottom of a rolling object equals zero

Torque and Rotational Dynamics

• Torque ($\tau$) is the rotational equivalent of force, causing an object to rotate about an axis
  • Calculated using $\tau = rF\sin\theta$, where $r$ is the distance from the axis of rotation, $F$ is the force, and $\theta$ is the angle between $r$ and $F$
• The net torque on an object is related to its angular acceleration by $\tau_{net} = I\alpha$
  • $I$ is the object's moment of inertia, a measure of its resistance to rotational acceleration
• Moment of inertia depends on the object's mass and its distribution relative to the axis of rotation
  • For a point mass, $I = mr^2$, where $m$ is the mass and $r$ is the distance from the axis
  • For extended objects, moment of inertia is calculated using integration or by using standard formulas (rod, disk, sphere)
• Newton's second law for rotational motion: The net torque on an object equals its moment of inertia times its angular acceleration

Rotational Energy

• Rotational kinetic energy ($K_r$) is the energy an object possesses due to its rotational motion
  • Calculated using $K_r = \frac{1}{2}I\omega^2$, where $I$ is the moment of inertia and $\omega$ is the angular velocity
• Total kinetic energy of an object combines translational and rotational kinetic energy ($K = K_t + K_r$)
• Work done by a torque is equal to the change in rotational kinetic energy
  • $W = \Delta K_r = \frac{1}{2}I(\omega_f^2 - \omega_i^2)$, where $\omega_i$ and $\omega_f$ are initial and final angular velocities
• Power in rotational motion is the rate at which work is done by a torque
  • $P = \tau\omega$, where $\tau$ is the torque and $\omega$ is the angular velocity

Angular Momentum

• Angular momentum ($L$) is the rotational analog of linear momentum
  • Calculated using $L = I\omega$, where $I$ is the moment of inertia and $\omega$ is the angular velocity
• The net external torque on a system equals the rate of change of its angular momentum ($\tau_{net} = \frac{dL}{dt}$)
  • If the net external torque is zero, angular momentum is conserved
• Angular impulse ($\Delta L$) is the change in angular momentum, equal to the product of torque and time interval
  • $\Delta L = \tau\Delta t$
• Conservation of angular momentum is crucial in understanding the behavior of rotating systems (figure skaters, planets, galaxies)

Conservation Laws in Rotation

• Energy is conserved in systems with both translational and rotational motion
  • Total energy ($E$) is the sum of kinetic (translational and rotational) and potential energy
  • $E = K_t + K_r + U = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 + mgh$, where $m$ is mass, $v$ is velocity, $I$ is moment of inertia, $\omega$ is angular velocity, $g$ is acceleration due to gravity, and $h$ is height
• In the absence of external torques, angular momentum is conserved
  • $L_i = L_f$ or $I_i\omega_i = I_f\omega_f$
• When a system's moment of inertia changes, its angular velocity must change to conserve angular momentum
  • Example: A figure skater increases their angular velocity by pulling their arms in, decreasing their moment of inertia

Real-World Applications

• Rolling motion: Wheels, cylinders, and spheres rolling down inclines or across surfaces
  • Combine translational and rotational motion, using conservation of energy and the relationship between linear and angular velocity
• Gears and pulleys: Used to transfer rotational motion and torque between objects
  • Gear ratios determine the relationship between angular velocities and torques of connected gears
• Flywheels: Used to store rotational kinetic energy and smooth out variations in angular velocity
  • Examples include potter's wheels and engines
• Gyroscopes: Utilize conservation of angular momentum to maintain orientation or measure angular velocity
  • Applications in navigation (gyrocompasses) and stabilization (Hubble Space Telescope)
• Planetary and stellar motion: Conservation of angular momentum plays a crucial role in the formation and evolution of planets, stars, and galaxies
  • Example: The Sun's rotation rate is slower than expected due to the transfer of angular momentum to the planets during the Solar System's formation