🌊College Physics II – Mechanics, Sound, Oscillations, and Waves Unit 10 – Fixed-Axis Rotation

Key Concepts and Definitions

• Fixed-axis rotation involves an object rotating about a fixed axis, with all points on the object moving in circular paths centered on the axis
• Angular displacement ($\Delta\theta$) measures the angle through which an object rotates, typically expressed in radians
• Angular velocity ($\omega$) describes the rate of change of angular displacement with respect to time, measured in radians per second
• Angular acceleration ($\alpha$) represents 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, and is equal to the product of the force and the perpendicular distance from the axis of rotation to the line of action of the force
• Moment of inertia ($I$) is a measure of an object's resistance to rotational motion, depending on the object's mass and its distribution relative to the axis of rotation
• Angular momentum ($L$) is the rotational equivalent of linear momentum, defined as the product of an object's moment of inertia and its angular velocity

Angular Motion Basics

• Angular motion describes the motion of an object rotating about a fixed axis, with the object's position defined by its angular displacement from a reference point
• The relationship between angular displacement ($\Delta\theta$), arc length ($s$), and radius ($r$) is given by $s = r\Delta\theta$
• Angular velocity ($\omega$) is defined as the rate of change of angular displacement with respect to time, expressed as $\omega = \frac{\Delta\theta}{\Delta t}$
  • Instantaneous angular velocity is given by $\omega = \lim_{\Delta t \to 0} \frac{\Delta\theta}{\Delta t} = \frac{d\theta}{dt}$
• Angular acceleration ($\alpha$) is the rate of change of angular velocity with respect to time, expressed as $\alpha = \frac{\Delta\omega}{\Delta t}$
  • Instantaneous angular acceleration is given by $\alpha = \lim_{\Delta t \to 0} \frac{\Delta\omega}{\Delta t} = \frac{d\omega}{dt}$
• The direction of angular velocity and angular acceleration vectors is determined by the right-hand rule, with the thumb pointing in the direction of the vector and the fingers curling in the direction of rotation
• Tangential velocity ($v_t$) is the linear velocity of a point on a rotating object, perpendicular to the radius, and is related to angular velocity by $v_t = r\omega$
• Centripetal acceleration ($a_c$) is the acceleration directed toward the center of the circular path, given by $a_c = \frac{v_t^2}{r} = r\omega^2$

Rotational Kinematics

• Rotational kinematics describes the motion of an object undergoing fixed-axis rotation, using equations analogous to those for linear motion
• The rotational kinematic equations for constant angular acceleration are:
  • $\Delta\theta = \omega_0 t + \frac{1}{2}\alpha t^2$
  • $\omega = \omega_0 + \alpha t$
  • $\omega^2 = \omega_0^2 + 2\alpha\Delta\theta$
• These equations relate angular displacement ($\Delta\theta$), initial angular velocity ($\omega_0$), final angular velocity ($\omega$), angular acceleration ($\alpha$), and time ($t$)
• To solve rotational kinematics problems, identify the known quantities and choose the appropriate equation to find the unknown variable
• When an object undergoes both rotational and translational motion, the linear and angular quantities can be related using the radius of rotation ($r$)
  • Linear displacement: $\Delta s = r\Delta\theta$
  • Linear velocity: $v = r\omega$
  • Linear acceleration: $a_t = r\alpha$
• Rotational motion problems often involve analyzing the motion of individual points on a rotating object, requiring the use of both linear and angular quantities

Torque and Rotational Dynamics

• Torque ($\tau$) 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 perpendicular distance from the axis of rotation to the line of action of the force $F$, and $\theta$ is the angle between the force and the radius vector
• The direction of torque is determined by the right-hand rule, with the thumb pointing in the direction of the axis of rotation and the fingers curling in the direction of the torque
• 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 equivalent of Newton's second law: $\sum\tau = I\alpha$
  • $I$ is the object's moment of inertia, and $\alpha$ is its angular acceleration
• Equilibrium in rotational dynamics occurs when the net torque acting on an object is zero ($\sum\tau = 0$), resulting in no angular acceleration
• The rotational equivalent of Newton's third law states that when two objects interact, they exert equal and opposite torques on each other
• Torque and angular acceleration are related to an object's moment of inertia, which depends on its mass distribution relative to the axis of rotation

Rotational Energy and Work

• Rotational kinetic energy ($K_r$) is the kinetic energy associated with an object's rotational motion, given by $K_r = \frac{1}{2}I\omega^2$, where $I$ is the object's moment of inertia and $\omega$ is its angular velocity
• Work done by a torque ($W_\tau$) is the product of the torque and the angular displacement through which it acts, expressed as $W_\tau = \tau\Delta\theta$
  • For a constant torque, the work done is $W_\tau = \tau\Delta\theta$
  • For a variable torque, the work done is $W_\tau = \int_{\theta_1}^{\theta_2} \tau(\theta) d\theta$
• The work-energy theorem for rotational motion states that the net work done by torques on an object equals the change in its rotational kinetic energy: $W_{\tau,net} = \Delta K_r = \frac{1}{2}I(\omega_2^2 - \omega_1^2)$
• Power in rotational motion ($P_r$) is the rate at which work is done by a torque, given by $P_r = \tau\omega$
• Conservation of energy applies to rotational motion, with the total energy (rotational kinetic energy plus potential energy) remaining constant in the absence of non-conservative forces
• In many real-world applications, an object experiences both translational and rotational motion, and its total kinetic energy is the sum of its translational and rotational kinetic energies: $K_{total} = K_{translational} + K_{rotational} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$

Moment of Inertia

• Moment of inertia ($I$) is a measure of an object's resistance to rotational motion, depending on its mass and the distribution of that mass relative to the axis of rotation
• The moment of inertia is given by $I = \int r^2 dm$, where $r$ is the perpendicular distance from the axis of rotation to the mass element $dm$
• For a discrete system of point masses, the moment of inertia is calculated as $I = \sum_{i} m_i r_i^2$, where $m_i$ is the mass of the $i$-th particle and $r_i$ is its distance from the axis of rotation
• The parallel-axis theorem states that the moment of inertia about any axis parallel to an axis through the object's center of mass is given by $I = I_{cm} + Md^2$, where $I_{cm}$ is the moment of inertia about the center of mass, $M$ is the total mass, and $d$ is the distance between the two axes
• The radius of gyration ($k$) is the distance from the axis of rotation at which all of the object's mass could be concentrated without changing its moment of inertia, given by $k = \sqrt{\frac{I}{M}}$
• Moments of inertia for common shapes (thin rod, thin rectangular plate, solid cylinder, thin-walled hollow cylinder, solid sphere, and thin-walled hollow sphere) can be calculated using specific formulas based on their dimensions and mass
• The moment of inertia plays a crucial role in determining an object's angular acceleration under the influence of a given torque, as seen in the rotational equivalent of Newton's second law: $\sum\tau = I\alpha$

Angular Momentum

• Angular momentum ($L$) is the rotational analog of linear momentum, defined as the product of an object's moment of inertia ($I$) and its angular velocity ($\omega$): $L = I\omega$
• The vector direction of angular momentum is determined by the right-hand rule, with the thumb pointing in the direction of the angular velocity vector
• The net external torque acting on an object equals the rate of change of its angular momentum, known as the angular momentum version of Newton's second law: $\sum\tau_{ext} = \frac{dL}{dt}$
• Conservation of angular momentum states that the total angular momentum of a system remains constant if no net external torque acts on the system: $L_{initial} = L_{final}$ or $I_1\omega_1 = I_2\omega_2$
  • This principle explains phenomena such as the increase in angular velocity when a spinning figure skater pulls their arms in, reducing their moment of inertia
• The angular impulse-momentum theorem states that the change in angular momentum of an object equals the angular impulse applied to it: $\Delta L = \int \tau dt$
• In the absence of external torques, the angular momentum of a system is conserved, which has important applications in astronomy (e.g., the formation and rotation of galaxies) and engineering (e.g., the design of flywheels and gyroscopes)
• When a system experiences a change in its moment of inertia, its angular velocity will change to conserve angular momentum, as seen in the equation $I_1\omega_1 = I_2\omega_2$

Applications and Real-World Examples

• Rotational motion is ubiquitous in everyday life, from the spinning of a CD or DVD to the rotation of a bicycle wheel or a car's tires
• Flywheels are devices that store rotational kinetic energy and are used in applications such as engines and power plants to smooth out fluctuations in power output
• Gyroscopes are devices that utilize the principles of angular momentum conservation to maintain their orientation, and are used in applications such as navigation systems, smartphones, and video game controllers
• The concept of torque is essential in the design and operation of many tools and machines, such as wrenches, screwdrivers, and gears
• In biomechanics, an understanding of torque and rotational motion is crucial for analyzing the movement of joints and limbs, as well as the forces exerted by muscles
• The moment of inertia plays a significant role in the design of structures and machines, as it determines an object's resistance to rotational motion and its response to applied torques
• Angular momentum conservation explains the motion of celestial bodies, such as the rotation of planets and the orbits of moons and satellites
• In fluid dynamics, the concepts of rotational motion are essential for understanding phenomena such as vortices and turbulence
• The principles of rotational energy and work are applied in the design of engines, turbines, and other rotational power sources
• Rotational kinematics is used in robotics and automation to control the motion of robotic arms, wheels, and other rotating components