Rotation of Rigid Bodies and Angular Kinematics

Fundamental Concepts of Rigid-Body Rotation

• Rigid Body Definition: A rigid body is an idealized model of an object that has a perfectly definite and unchanging shape and size. In this model, deformations such as stretching, twisting, and squeezing caused by external forces are ignored.
• Rotational Motion: This occurs when an object rotates about an axis that is stationary in an inertial frame of reference. Examples include airplane propellers, Blu-ray discs, Ferris wheels, and circular saw blades.
• Fixed-Axis Rotation: The simplest case of rotation involve a rigid body rotating around an axis that is at rest in an inertial frame and does not change direction relative to that frame. Examples include motor shafts, chunks of beef on a barbecue skewer, or a merry-go-round.

Angular Velocity and Acceleration

• Angular Coordinate ($\theta$): To describe the rotational position of a rigid body, a reference line $OP$ is chosen. The angle $\theta$ that this line makes with the positive $+x$-axis is the single coordinate needed.
  • Counterclockwise rotation is typically chosen as the positive direction.
  • Clockwise rotation is typically chosen as the negative direction.
• Radians: The most natural unit for measuring angles. One radian ($1\,rad$) is the angle subtended at the center of a circle by an arc length ($s$) equal to the radius ($r$) of the circle.
  • Formula for angle in radians: $\theta = \frac{s}{r}$
  • Relationship between arc length and radius: $s = r\theta$
  • Conversions: $1\,rad = \frac{360^{\circ}}{2\pi} \approx 57.3^{\circ}$; $180^{\circ} = \pi\,rad$.
  • There are $2\pi\,rad$ (approximately $6.283\,rad$) in one complete revolution ($360^{\circ}$).
• Angular Velocity ($\omega_z$): Defined as the rate of change of the angular coordinate.
  • Average Angular Velocity ($\omega_{av-z}$): The ratio of angular displacement $\Delta\theta$ to the time interval $\Delta t$: $\omega_{av-z} = \frac{\theta_2 - \theta_1}{t_2 - t_1} = \frac{\Delta\theta}{\Delta t}$.
  • Instantaneous Angular Velocity ($\omega_z$): The limit of average angular velocity as the time interval approaches zero: $\omega_z = \lim_{\Delta t \rightarrow 0} \frac{\Delta\theta}{\Delta t} = \frac{d\theta}{dt}$.
  • Units: Radians per second ($rad/s$) or revolutions per minute ($rev/min$ or $rpm$).
  • Conversion: $1\,rad/s \approx 10\,rpm$.
  • At any instant, every part of a rotating rigid body has the same angular velocity.
• Angular Velocity as a Vector ($\vec{\Omega}$): Directed along the axis of rotation. Its direction is determined by the right-hand rule:
  • If you curl the fingers of your right hand in the direction of rotation, the thumb points in the direction of the angular velocity vector.
  • The vector is perpendicular to the plane of rotation, never in it.
• Angular Acceleration ($\alpha_z$): The rate of change of angular velocity.
  • Average Angular Acceleration ($\alpha_{av-z}$): $\alpha_{av-z} = \frac{\omega_{2z} - \omega_{1z}}{t_2 - t_1} = \frac{\Delta\omega_z}{\Delta t}$.
  • Instantaneous Angular Acceleration ($\alpha_z$): $\alpha_z = \lim_{\Delta t \rightarrow 0} \frac{\Delta\omega_z}{\Delta t} = \frac{d\omega_z}{dt} = \frac{d^2\theta}{dt^2}$.
  • Units: Radians per second per second ($rad/s^2$).
  • If $\alpha_z$ and $\omega_z$ have the same sign, the body is speeding up; if they have opposite signs, it is slowing down.

Rotation with Constant Angular Acceleration

• When angular acceleration $\alpha_z$ is constant, kinematic relationships are analogous to those for straight-line motion:
  • $\omega_z = \omega_{0z} + \alpha_z t$
  • $\theta = \theta_0 + \omega_{0z}t + \frac{1}{2}\alpha_z t^2$
  • $\omega_z^2 = \omega_{0z}^2 + 2\alpha_z(\theta - \theta_0)$
  • $\theta - \theta_0 = \frac{1}{2}(\omega_{0z} + \omega_z)t$
• Comparison of Linear vs. Angular Equations:
  • Position: $x \leftrightarrow \theta$
  • Velocity: $v_x \leftrightarrow \omega_z$
  • Acceleration: $a_x \leftrightarrow \alpha_z$

Relationships Between Linear and Angular Kinematics

• Linear Speed ($v$): A point at a distance $r$ from the rotation axis has a linear speed proportional to the body's angular speed $\omega$: $v = r\omega$. This point travels along a circular path of radius $r$. This formula only holds when $\omega$ is in $rad/s$.
• Linear Acceleration ($\vec{a}$): A particle in a rotating body has two components of acceleration:
  • Tangential Component ($a_{tan}$): Changes the magnitude of the velocity (the speed). It is equal to the rate of change of speed: $a_{tan} = \frac{dv}{dt} = r\frac{d\omega}{dt} = r\alpha$.
  • Centripetal (Radial) Component ($a_{rad}$): Changes the direction of the velocity. It always points toward the axis of rotation: $a_{rad} = \frac{v^2}{r} = \omega^2 r$.
  • Resultant Acceleration Magnitude ($a$): Since the tangential and radial components are perpendicular, the magnitude is: $a = \sqrt{a_{tan}^2 + a_{rad}^2}$.

Rotational Kinetic Energy and Moment of Inertia

• Rotational Kinetic Energy ($K$): The kinetic energy of a rotating rigid body is the sum of the kinetic energies of all its constituent particles.
  • $K = \frac{1}{2}I\omega^2$
  • To use this formula, $\omega$ must be in radians per second to yield energy in Joules.
• Moment of Inertia ($I$): A measure of a body's rotational inertia (resistance to changes in rotation).
  • For a collection of point masses: $I = m_1 r_1^2 + m_2 r_2^2 + \dots = \sum_{i} m_i r_i^2$
  • SI Unit: Kilogram-meter$^2$ ($kg \cdot m^2$).
  • The moment of inertia depends on the location and orientation of the axis. It is not a fixed property like total mass.
• Comparison to Linear Motion: Moment of inertia $I$ is the rotational analog of mass $m$. While mass is a measure of an object's resistance to translational acceleration, moment of inertia is a measure of resistance to angular acceleration.
• Gravitational Potential Energy ($U$): For an extended body (rigid or not), if gravity $g$ is uniform, the potential energy is calculated as if all mass were at the center of mass: $U = Mgy_{cm}$.

Moments of Inertia for Common Shapes (Uniform Density)

• Slender rod, axis through center ($L =$ length): $I = \frac{1}{12}ML^2$
• Slender rod, axis through one end: $I = \frac{1}{3}ML^2$
• Rectangular plate, axis through center ($a, b =$ sides): $I = \frac{1}{12}M(a^2 + b^2)$
• Thin rectangular plate, axis along edge ($a =$ side perpendicular to axis): $I = \frac{1}{3}Ma^2$
• Hollow cylinder (inner radius $R_1$, outer radius $R_2$): $I = \frac{1}{2}M(R_1^2 + R_2^2)$
• Solid cylinder (radius $R$): $I = \frac{1}{2}MR^2$
• Thin-walled hollow cylinder (radius $R$): $I = MR^2$
• Solid sphere (radius $R$): $I = \frac{2}{5}MR^2$
• Thin-walled hollow sphere (radius $R$): $I = \frac{2}{3}MR^2$

Parallel-Axis Theorem

• Theorem Statement: There is a simple relationship between the moment of inertia ($I_{cm}$) of a body about an axis through its center of mass and the moment of inertia ($I_P$) about any other axis parallel to the original axis.
• Formula: $I_P = I_{cm} + Md^2$
  • $M$: Total mass of the body.
  • $d$: Perpendicular distance between the two parallel axes.
• Implication: A rigid body has its minimum possible moment of inertia about an axis passing through its center of mass. It is easier to start a body rotating if the axis passes through the center of mass.

Moment-of-Inertia Calculations via Integration

• General Integral: For continuous mass distributions, the sum becomes an integral over the mass elements ($dm$): $I = \int r^2 \,dm$.
• Using Density ($\rho$): If density is constant, $dm = \rho \,dV$, where $dV$ is a volume element.
  • $I = \rho \int r^2 \,dV$
• Example: Hollow Cylinder: Dividing the cylinder into thin cylindrical shells of radius $r$, thickness $dr$, and length $L$:
  • $dm = \rho(2\pi rL \,dr)$
  • $I = \int_{R1}^{R2} r^2(\rho 2\pi rL \,dr) = \frac{\pi\rho L}{2}(R_2^4 - R_1^4) = \frac{1}{2}M(R_1^2 + R_2^2)$.
• Example: Uniform Sphere: Dividing the sphere into thin solid disks of thickness $dx$ and radius $y = \sqrt{R^2 - x^2}$. The moment of inertia for each disk is $dI = \frac{1}{2}y^2 \,dm$.
  • Integrating from $x = -R$ to $x = R$ yields: $I = \frac{2}{5}MR^2$.

Biological and Scientific Applications

• E. coli Bacteria: These organisms swim by rotating corkscrew-shaped flagella. Their protein motors can rotate flagella at angular speeds of $200$ to $1000\,rev/min$ ($20$ to $100\,rad/s$) and provide angular acceleration to vary speed.
• Bird Flight: A hummingbirds has small wings with low moments of inertia, allowing flap rates up to $70$ beats per second. The Andean condor (Vultur gryphus) has immense wings with large moments of inertia, flapping at only about $1$ beat per second during takeoff and preferring to soar.
• Earth's Moment of Inertia: Measured via satellite orbit variations. This reveals that the Earth is far denser at its core than in its outer layers.
• Neutron Stars/Crab Nebula: The Crab Nebula is a remnant of a supernova from $1054\,A.D.$ It releases energy at a rate of $5 \times 10^{31}\,W$ ($10^5$ times the sun's radiation). This energy comes from the rotational kinetic energy of a spinning neutron star cooling at its center, which rotates once every $0.0331\,s$.
• Human Rotational Energy: Evaluation of a dancer's spinning involves modeling body segments (head, arms, trunk, and legs). Arms contribute significantly to the moment of inertia when outstretched despite being only $13\%$ of total body mass.

Questions & Discussion

• Propeller Length Change: If an airplane propeller blade (modeled as a thin rod) is stretched to double its length while mass and angular speed remain constant, by what factor does kinetic energy increase?
  • Response: Since $I_{rod} = \frac{1}{3}ML^2$, doubling $L$ increases $I$ by a factor of $2^2 = 4$. Because $K = \frac{1}{2}I\omega^2$, the kinetic energy increases by a factor of 4.
• Blu-ray Player Scanning: As the scanning head moves outward on a Blu-ray disc track (larger radius $r$), how must the rotation speed $\omega$ change to maintain a constant linear scanning speed $v$?
  • Response: Since $v = r\omega$, if $v$ is constant and $r$ increases, $\omega$ must decrease.
• Pool Cue Moment of Inertia: Does a pool cue have a larger moment of inertia for an axis perpendicular to its length through the thicker end or the thinner end?
  • Response: More mass is concentrated at the thicker end, so the center of mass is closer to that end. According to the parallel-axis theorem ($I_P = I_{cm} + Md^2$), the thinner end, being further from the center of mass (larger $d$), yields a larger moment of inertia.