🔋College Physics I – Introduction Unit 10 – Rotational Motion & Angular Momentum

Key Concepts and Definitions

• Angular displacement ($\theta$) measures the angle through which an object rotates about an axis
• Angular velocity ($\omega$) represents the rate of change of angular displacement with respect to time ($\omega = \frac{d\theta}{dt}$)
  • Measured in radians per second (rad/s)
  • Can be constant or varying
• Angular acceleration ($\alpha$) describes the rate of change of angular velocity with respect to time ($\alpha = \frac{d\omega}{dt}$)
  • Measured in radians per second squared (rad/s²)
  • Indicates how quickly an object's angular velocity changes
• Moment of inertia ($I$) quantifies an object's resistance to rotational motion
  • Depends on the object's mass distribution and shape
  • Calculated using the formula $I = \sum mr^2$ for a system of particles or $I = \int r^2dm$ for a continuous mass distribution
• Torque ($\tau$) is the rotational equivalent of force, causing an object to rotate about an axis
  • Defined as the cross product of the position vector and the force vector ($\tau = \vec{r} \times \vec{F}$)
  • Measured in newton-meters (N·m)

Angular Kinematics

• Angular kinematics describes the motion of objects undergoing rotational motion without considering the causes of the motion
• Angular displacement ($\Delta\theta$) is the angle through which an object rotates, measured in radians
  • Calculated using the formula $\Delta\theta = \theta_f - \theta_i$, where $\theta_f$ is the final angular position and $\theta_i$ is the initial angular position
• Angular velocity ($\omega$) represents the rate of change of angular displacement with respect to time
  • Average angular velocity is calculated using $\omega_{avg} = \frac{\Delta\theta}{\Delta t}$
  • Instantaneous angular velocity is given by $\omega = \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t} = \frac{d\theta}{dt}$
• Angular acceleration ($\alpha$) describes the rate of change of angular velocity with respect to time
  • Average angular acceleration is calculated using $\alpha_{avg} = \frac{\Delta\omega}{\Delta t}$
  • Instantaneous angular acceleration is given by $\alpha = \lim_{\Delta t \to 0} \frac{\Delta\omega}{\Delta t} = \frac{d\omega}{dt}$
• Kinematic equations for rotational motion (assuming constant angular acceleration):
  • $\theta_f = \theta_i + \omega_i t + \frac{1}{2}\alpha t^2$
  • $\omega_f = \omega_i + \alpha t$
  • $\omega_f^2 = \omega_i^2 + 2\alpha(\theta_f - \theta_i)$

Torque and Rotational Equilibrium

• Torque ($\tau$) is the rotational equivalent of force, causing an object to rotate about an axis
  • Calculated using the formula $\tau = rF\sin\theta$, where $r$ is the distance from the axis of rotation to the point of force application, $F$ is the magnitude of the force, and $\theta$ is the angle between the force and the position vector
• Net torque ($\sum \tau$) is the sum of all torques acting on an object
  • Positive torques cause counterclockwise rotation, while negative torques cause clockwise rotation
• Rotational equilibrium occurs when the net torque acting on an object is zero ($\sum \tau = 0$)
  • In rotational equilibrium, the object either remains at rest or rotates with constant angular velocity
• Center of mass is the point at which an object's mass can be considered to be concentrated for the purpose of analyzing its motion
  • For objects with uniform density, the center of mass coincides with the geometric center
• Couple is a pair of equal and opposite forces that cause rotation without translation
  • The torque produced by a couple is independent of the choice of the axis of rotation

Rotational Dynamics and Newton's Second Law

• Newton's second law for rotational motion states that the net torque acting on an object is equal to the product of its moment of inertia and angular acceleration ($\sum \tau = I\alpha$)
  • This is analogous to $\sum F = ma$ for translational motion
• Moment of inertia ($I$) is a measure of an object's resistance to rotational motion
  • Depends on the object's mass distribution and shape
  • 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, $I = \int r^2dm$, where $dm$ is an infinitesimal mass element
• Parallel-axis theorem states that the moment of inertia about any axis parallel to an axis through the center of mass is equal to the moment of inertia about the axis through the center of mass plus the product of the total mass and the square of the perpendicular distance between the axes ($I = I_{CM} + Md^2$)
• Angular acceleration ($\alpha$) is directly proportional to the net torque and inversely proportional to the moment of inertia
  • Increasing the net torque or decreasing the moment of inertia results in a higher angular acceleration

Rotational Energy and Work

• Rotational kinetic energy ($K_r$) is the energy associated with an object's rotational motion
  • Calculated using the formula $K_r = \frac{1}{2}I\omega^2$, where $I$ is the moment of inertia and $\omega$ is the angular velocity
• Work done by a torque ($W_\tau$) is the product of the torque and the angular displacement ($W_\tau = \tau\Delta\theta$)
  • Positive work is done when the torque acts in the same direction as the angular displacement
  • Negative work is done when the torque acts in the opposite direction of the angular displacement
• Work-energy theorem for rotational motion states that the net work done by external torques on an object is equal to the change in its rotational kinetic energy ($W_{net} = \Delta K_r$)
  • This is analogous to the work-energy theorem for translational motion ($W_{net} = \Delta K$)
• Power in rotational motion ($P_r$) is the rate at which work is done by a torque
  • Calculated using the formula $P_r = \tau\omega$, where $\tau$ is the torque and $\omega$ is the angular velocity
• Conservation of energy applies to rotational motion, considering both rotational kinetic energy and potential energy (gravitational, elastic, etc.)

Angular Momentum and Conservation

• Angular momentum ($\vec{L}$) is a vector quantity that represents the rotational analog of linear momentum
  • Defined as the cross product of the position vector and the linear momentum vector ($\vec{L} = \vec{r} \times \vec{p}$)
  • For a point mass, $\vec{L} = m\vec{r} \times \vec{v}$, where $m$ is the mass, $\vec{r}$ is the position vector, and $\vec{v}$ is the velocity vector
  • For an object rotating about a fixed axis, $L = I\omega$, where $I$ is the moment of inertia and $\omega$ is the angular velocity
• Conservation of angular momentum states that the total angular momentum of a system remains constant if no external torques act on the system ($\Delta\vec{L} = 0$ when $\sum \vec{\tau}_{ext} = 0$)
  • This is analogous to the conservation of linear momentum in the absence of external forces
• Angular impulse ($\Delta\vec{L}$) is the change in angular momentum caused by a torque acting over a period of time
  • Calculated using the formula $\Delta\vec{L} = \int \vec{\tau} dt$, where $\vec{\tau}$ is the torque and $dt$ is an infinitesimal time interval
• Precession is the gradual change in the orientation of a rotating object's axis of rotation when subjected to an external torque
  • Examples include the precession of a spinning top and the precession of the Earth's rotational axis (which causes the phenomenon of the precession of the equinoxes)

Applications and Real-World Examples

• Flywheel is a rotating mechanical device that stores rotational kinetic energy and helps to smooth out variations in angular velocity
  • Used in engines, machines, and power plants to regulate rotational motion and maintain a constant angular velocity
• Gyroscope is a device consisting of a spinning wheel or disc mounted on a gimbal, which allows it to maintain its orientation regardless of external torques
  • Used in navigation systems, inertial guidance systems, and stabilization devices (e.g., Hubble Space Telescope, smartphones, and virtual reality headsets)
• Centripetal force is the force that causes an object to follow a curved path and is always directed toward the center of curvature
  • In rotational motion, the centripetal force is provided by the tension in the string, the normal force from the surface, or the gravitational force
• Centrifugal force is a fictitious force that appears to act on an object moving in a circular path, pushing it away from the center of rotation
  • It is an inertial effect and is not a real force, but it can be used to simplify the analysis of rotational motion in a rotating reference frame
• Banked curves on roads and racetracks are designed to provide the necessary centripetal force for vehicles to maintain a circular path without relying solely on friction
  • The angle of the bank depends on the desired speed of the vehicles and the radius of curvature of the track

Problem-Solving Strategies

• Identify the type of rotational motion (constant angular velocity, constant angular acceleration, or varying angular acceleration)
• Draw a clear diagram of the problem, labeling all relevant quantities (forces, torques, distances, angles, etc.)
• Determine the axis of rotation and the moment of inertia of the object(s) involved
  • Use the parallel-axis theorem if necessary
• Apply the appropriate equations and principles based on the given information and the quantities to be found
  • Use the rotational kinematic equations for problems involving angular displacement, velocity, and acceleration
  • Use Newton's second law for rotational motion ($\sum \tau = I\alpha$) for problems involving torques and angular acceleration
  • Use the work-energy theorem for rotational motion ($W_{net} = \Delta K_r$) for problems involving work and rotational kinetic energy
  • Use the conservation of angular momentum ($\Delta\vec{L} = 0$ when $\sum \vec{\tau}_{ext} = 0$) for problems involving isolated systems and angular momentum
• Check the units of your answer to ensure they are consistent with the quantity being solved for
• Verify that your answer makes sense in the context of the problem and the real world