PHD
  1. Classical mechanics  1.1. Newtonian mechanics  1.1.1. Laws of motion in Newtonian mechanics  1.1.2. Dynamics of particles and systems  1.1.3. Conservation laws  1.1.4. Central force motion  1.2. Lagrangian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Constraints and normalized coordinates  1.2.4. Noether's theorem  1.3. Hamiltonian mechanics  1.3.1. Hamilton's equations  1.3.2. Canonical conversion  1.3.3. Poisson bracket  1.3.4. Action-angle variable  1.4. Rigid body mobility  1.4.1. Rotation of rigid bodies  1.4.2. Moment of inertia tensor  1.4.3. Euler's equations in rigid body dynamics  1.4.4. Gyroscopic motion  1.5. Chaos and nonlinear dynamics  1.5.1. Stage space and attractive  1.5.2. Bifurcation and chaos theory  1.5.3. Hamiltonian Chaos  1.5.4. Lyapunov exponent
  2. Electrodynamics  2.1. Maxwell's equations  2.1.1. Gauss's law for electricity  2.1.2. Gauss's law for magnetism  2.1.3. Faraday's Law  2.1.4. Ampere's Law  2.1.5. Boundary conditions in Maxwell's equations  2.2. Electromagnetic waves  2.2.1. Wave equation  2.2.2. Polarization  2.2.3. Reflection and Refraction  2.2.4. Waveguides and resonators  2.3. Special relativity  2.3.1. Lorentz transformations  2.3.2. Relativistic energy and momentum  2.3.3. Relativistic electrodynamics  2.3.4. Minkowski spacetime  2.4. Radiation and scattering  2.4.1. Dipole radiation  2.4.2. Multipole expansion  2.4.3. Compton scattering  2.4.4. Thomson and Rayleigh scattering  2.5. Plasma Physics  2.5.1. Debye Screening  2.5.2. Magnetohydrodynamics  2.5.3. Plasma instability  2.5.4. Fusion Plasma
  3. Quantum mechanics  3.1. Foundations of quantum mechanics  3.1.1. Principles of quantum mechanics  3.1.2. Wave function and probability interpretation  3.1.3. Uncertainty principle  3.1.4. Quantum Tunneling  3.2. Schrödinger Equation  3.2.1. Time-dependent Schrödinger equation  3.2.2. Time-independent Schrödinger equation  3.2.3. Eigenvalues and eigenfunctions in the Schrödinger equation  3.2.4. Path integrals in quantum mechanics  3.3. Quantum operators  3.3.1. Commutators and observables  3.3.2. Angular momentum operator  3.3.3. Ladder Operator  3.3.4. Pauli matrices  3.4. Quantum entanglement and measurement  3.4.1. Bell's theorem  3.4.2. Quantum Teleportation  3.4.3. Quantum entanglement and quantum decoherence in measurement  3.4.4. Quantum entanglement and measurement in quantum mechanics  3.5. Relativistic quantum mechanics  3.5.1. Klein–Gordon equation  3.5.2. Dirac equation  3.5.3. Quantum field theory  3.5.4. Feynman path integral
  4. Statistical mechanics and thermodynamics  4.1. Classical thermodynamics  4.1.1. Laws of Thermodynamics  4.1.2. Carnot cycle  4.1.3. Entropy and free energy  4.1.4. Thermodynamic efficiency  4.2. Kinetic theory of gases  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Transport phenomenon  4.2.3. Mean free path  4.2.4. Non-equilibrium systems in the kinetic theory of gases  4.3. Statistical mechanics  4.3.1. Microstates and Macrostates  4.3.2. Partition function  4.3.3. Bose–Einstein and Fermi–Dirac statistics  4.3.4. Fluctuations and correlations  4.4. Phase transition  4.4.1. Important events  4.4.2. Landau theory  4.4.3. Ising model  4.4.4. Renormalization group theory
  5. Quantum field theory  5.1. Second Quantization  5.1.1. Quantum harmonic oscillator in second quantization  5.1.2. Creation and annihilation operators in second quantization  5.1.3. Path Integral in Second Quantization  5.1.4. Fock space  5.2. Quantum Electrodynamics  5.2.1. Feynman diagrams  5.2.2. Renormalization in quantum electrodynamics  5.2.3. Gauge invariance in quantum electrodynamics  5.2.4. Vacuum polarization  5.3. Quantum Chromodynamics  5.3.1. Quarks and gluons  5.3.2. Color range  5.3.3. Lattice QCD  5.3.4. Asymptotic freedom  5.4. Standard model of particle physics  5.4.1. Electroweak theory  5.4.2. Higgs mechanism  5.4.4. Grand Unified Theories
  6. General relativity and gravity  6.1. Einstein's field equations  6.1.1. Ricci tensor and scalar curvature  6.1.2. Schwarzschild solution  6.1.3. Kerr metric  6.1.4. Gravitational waves  6.2. Cosmology  6.2.1. Friedmann equation  6.2.2. Cosmic inflation  6.2.3. Dark matter and dark energy  6.2.4. Large scale structure  6.3. Black holes and wormholes  6.3.1. Event horizon in black holes and wormholes  6.3.2. Hawking radiation  6.3.3. Penrose process  6.3.4. Information paradox in black holes and wormholes
  7. Condensed matter physics  7.1. Crystal structure and lattice  7.1.1. Bravais lattices  7.1.2. Band theory in crystal structure and lattices  7.1.3. Phonons in crystal structure and lattice  7.1.4. Block's theorem  7.2. Superconductivity  7.2.1. BCS principle  7.2.2. Meissner effect  7.2.3. High Temperature Superconductor  7.2.4. Josephson effect

PHD → Classical mechanics → Rigid body mobility ↓


Rotation of rigid bodies


In classical mechanics, the study of rotations of rigid bodies is an in-depth field that requires an understanding of several key concepts such as angular velocity, angular momentum, torque, and more. Rigid body dynamics is a branch of mechanics that deals with the motion of rigid bodies, which are objects that do not deform under the action of forces. The purpose of this lesson is to delve deeper into these concepts and provide a comprehensive exploration of the rotation of rigid bodies.

Defining a rigid body

A rigid body is an idealization of a solid body in which deformation is neglected. This means that the distance between any two given points on the body remains constant regardless of external forces or moments. Think of objects such as a spinning top, a rotating wheel, or even the Earth rotating on its axis. All of these can be treated as rigid bodies in the context of rotational dynamics.

Basic concepts of rotation

Angular displacement

Just like linear motion, we first introduce angular displacement for rotational motion. Angular displacement measures the angle through which a point, line, or body rotates in a specified sense about a specified axis. It is usually measured in radians.

θ = s / r

Where θ is the angular displacement, s is the arc length, and r is the radius of the circle.

Angular velocity

Angular velocity is a measure of the rate of change of angular displacement. It is a vector quantity and is measured in radians per second. If θ is the angular displacement and t is the time taken, then the angular velocity ω is given by:

ω = dθ / dt

In vector form, the direction of ω is along the axis of rotation, and it follows the right-hand rule.

Angular acceleration

Angular acceleration is the rate of change of angular velocity. It is represented by α and is measured in radians per second squared.

α = dω / dt

Analogous to linear acceleration, angular acceleration is the second derivative of angular displacement with respect to time.

Torque

Torque, often represented by the Greek letter τ, is a measure of rotational force. It is calculated by the cross product of the lever arm distance vector and the force vector:

τ = rx F

Where r is the position vector (distance from the axis of rotation), F is the force vector, and x represents the cross product. The direction of τ is perpendicular to the plane formed by r and F

Moment of inertia

The moment of inertia, often denoted by I, is the rotational analogue of mass in linear motion. It measures the extent to which an object resists angular acceleration. It depends on the mass distribution of the object relative to the axis of rotation. Mathematically, it is given as:

I = Σ m_i r_i^2

where m_i is the mass of a point located at a radius r_i from the axis of rotation. For continuous bodies, it is calculated using an integral:

I = ∫ r^2 dm

The calculation of the moment of inertia varies depending on the axis of rotation and the geometry of the object. Some standard forms are important in applications:

  • Solid cylinder or disc around its axis: I = (1/2) mr^2
  • Thin rod about an axis passing through the centre: I = (1/12) m L^2, where L is the length of the rod.

Angular momentum

Angular momentum is a vector quantity associated with the rotational motion of rigid bodies. Angular momentum, represented by L is the product of moment of inertia and angular velocity:

L = I ω

For a system of particles, the angular momentum is the sum of the angular momenta of all the individual particles. Conservation of angular momentum is a fundamental concept that states that in the absence of external torques, the angular momentum of a system remains constant.

Equations of motion for rotational dynamics

The fundamental equation corresponding to Newton's second law of rotation is:

τ = I α

This equation states that the torque applied to a rigid body is equal to the product of the moment of inertia and the angular acceleration produced by it. It provides the basis for the analysis of rotational systems.

Suppose a wheel is being rotated by a torque. To calculate its angular acceleration, rearrange the equation as follows:

α = τ / I

If more torque is applied the wheel will accelerate faster if the moment of inertia remains constant.

Work and energy in rotational motion

The work done on a rigid body in rotational motion can be described in terms of torque and angular displacement. The work-energy principle for rotational motion is given as:

W = τ θ

Where W is work, and θ is angular displacement in radians. The rotational kinetic energy of a body is expressed as:

KE_rotational = (1/2) I ω^2

This equation is parallel to the translational kinetic energy formula ((1/2) mv^2).

Gyroscopic motion

A gyroscope demonstrates the principles of rotation around its axis while being supported at a point. Its application relies on the conservation of angular momentum. Consider a wheel rotating on an axis; if the axis is tilted, the wheel can still maintain its orientation. This ability arises from the gyroscopic effect, which is important in technologies such as navigation.

Illustration of gyroscopic motion:

  • The circle represents the path of rotation
  • The blue line is the axle of the wheel
  • The red line shows the change in tilt or orientation

Precession and nutation

When an external torque such as gravity is applied to a top, the axis rotates slowly, this motion is known as precession. The rate of precession depends on the angular speed and the applied torque. If the top exhibits excess buoyancy, it is said to be nutated.

The precession angular velocity Ω depends on the torque and angular momentum:

Ω = τ / L

Applications of rigid body rotation

The principles of rigid body rotation are fundamental in engineering and everyday applications. Examples include:

  • Automobile engines: The rotating crankshaft and other components are designed based on rotational dynamics.
  • Aerodynamics: The rotation of an aircraft's turbines and propellers is critical to flight.
  • Merry-Go-Round: Provide an intuitive understanding of centrifugal forces and rotational motion.
  • Hard drives: Use rigid body rotation to read and write to a storage disk.

The rotation motion of celestial bodies such as planets and stars is also governed by these laws, extending its application to the astrophysical scale.

Conclusion

The study of the rotational dynamics of rigid bodies plays a vital role in understanding the physical world. The interactions between torque, moment of inertia, angular momentum, and energy provide insight into both natural phenomena and engineered devices. Mastery of these concepts equips individuals with the analytical tools necessary for problem-solving in physics and allied disciplines.


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Rotation of rigid bodies

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