🍏Principles of Physics I Unit 9 – Rotational Motion: Kinematics and Dynamics

Key Concepts and Definitions

• Angular displacement ($\theta$) measures the angle through which an object rotates in radians or degrees
• Angular velocity ($\omega$) represents the rate of change of angular displacement with respect to time, typically expressed in radians per second or revolutions per minute
• Angular acceleration ($\alpha$) describes the rate of change of angular velocity with respect to time, measured in radians per second squared
• Torque ($\tau$) is the rotational equivalent of force, causing an object to rotate about an axis, calculated as the product of force and the perpendicular distance from the axis of rotation (moment arm)
• Moment of inertia ($I$) quantifies an object's resistance to rotational motion, depending on its mass distribution relative to the axis of rotation
• Angular momentum ($L$) is the rotational analog of linear momentum, defined as the product of moment of inertia and angular velocity ($L = I\omega$)
• Rotational kinetic energy ($K_r$) is the energy associated with an object's rotational motion, calculated as $\frac{1}{2}I\omega^2$

Angular Motion Basics

• Rotational motion involves an object rotating about a fixed axis, with each point on the object moving in a circular path
• The axis of rotation is the line about which the object rotates, perpendicular to the plane of rotation
• Angular displacement is measured in radians, where one radian is the angle subtended by an arc length equal to the radius of the circle
  • To convert between radians and degrees, use the relationship: $2\pi$ radians = 360 degrees
• Tangential velocity ($v_t$) is the linear velocity of a point on a rotating object, perpendicular to the radius and related to angular velocity by $v_t = r\omega$
• Centripetal acceleration ($a_c$) is the acceleration directed towards the center of the circular path, given by $a_c = \frac{v_t^2}{r} = r\omega^2$
• Rotational motion can be either clockwise (CW) or counterclockwise (CCW), with the direction of angular velocity and acceleration defined by the right-hand rule

Rotational Kinematics Equations

• Rotational kinematics equations describe the relationships between angular displacement, velocity, acceleration, and time for objects undergoing rotational motion with constant angular acceleration
• The four main rotational kinematics equations are:
  1. $\theta = \theta_0 + \omega_0t + \frac{1}{2}\alpha t^2$
  2. $\omega = \omega_0 + \alpha t$
  3. $\theta = \theta_0 + \frac{1}{2}(\omega_0 + \omega)t$
  4. $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$
• These equations are analogous to the linear kinematics equations, with angular displacement, velocity, and acceleration replacing their linear counterparts
• To solve problems involving rotational motion, identify the known variables and choose the appropriate equation to find the unknown quantity
• When an object undergoes both linear and rotational motion (rolling without slipping), the linear and angular velocities are related by $v = r\omega$

Torque and Rotational Dynamics

• Torque is the rotational equivalent of force, causing an object to rotate about an axis
• The magnitude of torque is given by $\tau = rF\sin\theta$, where $r$ is the distance from the axis of rotation to the point where the force is applied (moment arm), $F$ is the magnitude of the force, and $\theta$ is the angle between the force and the moment arm
  • When the force is perpendicular to the moment arm, $\sin\theta = 1$, and the torque is simply $\tau = rF$
• The direction of torque is determined by the right-hand rule: curl your fingers in the direction of rotation, and your thumb points in the direction of the torque vector
• Net torque ($\sum\tau$) is the sum of all torques acting on an object, and it determines the object's angular acceleration according to the rotational version of Newton's second law: $\sum\tau = I\alpha$
• Equilibrium in rotational dynamics occurs when the net torque acting on an object is zero, resulting in no angular acceleration
  • Static equilibrium: the object is at rest and has no angular velocity or acceleration
  • Dynamic equilibrium: the object has a constant angular velocity and zero angular acceleration

Moment of Inertia

• Moment of inertia is a measure of an object's resistance to rotational motion, depending on its mass distribution relative to the axis of rotation
• The moment of inertia is calculated by integrating the product of the mass element ($dm$) and the square of its distance from the axis of rotation ($r^2$) over the entire object: $I = \int r^2dm$
• For objects with simple geometries and uniform mass distribution, the moment of inertia can be calculated using specific formulas:
  • Thin rod (length $L$, mass $M$, about an axis perpendicular to the rod at one end): $I = \frac{1}{3}ML^2$
  • Thin rectangular plate (width $w$, height $h$, mass $M$, about an axis perpendicular to the plate at its center): $I = \frac{1}{12}M(w^2 + h^2)$
  • Solid cylinder or disk (radius $R$, mass $M$, about its central axis): $I = \frac{1}{2}MR^2$
• The 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 mass and the square of the perpendicular distance between the axes: $I = I_{CM} + Md^2$

Angular Momentum and Conservation

• Angular momentum is the rotational analog of linear momentum, defined as the product of moment of inertia and angular velocity: $L = I\omega$
• The direction of angular momentum is determined by the right-hand rule: curl your fingers in the direction of rotation, and your thumb points in the direction of the angular momentum vector
• The law of conservation of angular momentum states that the total angular momentum of a system remains constant if no external torques act on the system: $L_i = L_f$ or $I_i\omega_i = I_f\omega_f$
  • This law explains phenomena such as the increase in angular velocity when a spinning figure skater pulls their arms in, reducing their moment of inertia
• Angular impulse ($\Delta L$) is the change in angular momentum, equal to the product of the net torque and the time interval over which it acts: $\Delta L = \sum\tau\Delta t$
• In the absence of external torques, the angular momentum of a system is conserved, allowing for the analysis of complex rotational motion problems

Real-World Applications

• Rotational motion is prevalent in various real-world scenarios, such as the motion of wheels, gears, and turbines
• Flywheels are devices that store rotational kinetic energy and help maintain a constant angular velocity in machines (engines, motors)
• Gyroscopes are devices that maintain their orientation due to conservation of angular momentum, used in navigation systems (airplanes, spacecraft) and stabilization applications (Segways, cameras)
• The concept of torque is essential in the design of tools and machines that involve rotation (wrenches, levers, gears)
• Understanding rotational dynamics is crucial for analyzing the motion of celestial bodies (planets, moons) and artificial satellites orbiting Earth
• In biomechanics, rotational motion principles are applied to study the movement of joints and limbs (elbows, knees) and the efficiency of human motion in sports (gymnastics, diving)

Problem-Solving Strategies

• Identify the type of rotational motion problem: constant angular velocity, constant angular acceleration, or conservation of angular momentum
• Draw a clear diagram of the system, labeling relevant variables such as forces, distances, and angles
• Determine the axis of rotation and the direction of angular quantities (displacement, velocity, acceleration, momentum) using the right-hand rule
• List the known quantities and the unknown variable to be solved for
• Select the appropriate equations based on the given information and the quantity to be found
  • For constant angular acceleration problems, use the rotational kinematics equations
  • For problems involving torque and angular acceleration, use the rotational version of Newton's second law: $\sum\tau = I\alpha$
  • For conservation of angular momentum problems, use the equation $I_i\omega_i = I_f\omega_f$
• Substitute the known values into the chosen equation and solve for the unknown variable
• Check the units of the solution and ensure that the answer is reasonable within the context of the problem