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


Rigid body dynamics is a fundamental topic in classical mechanics that deals with the motion of rigid bodies, which are solid objects that do not deform under the action of forces. Understanding how such bodies move is important to many fields, including engineering, robotics, biomechanics, and astrophysics.

A rigid body is defined as an object with a fixed shape that does not change despite the forces acting upon it. Unlike a point mass or particle that can only move linearly, a rigid body can move and rotate. This unique combination of motion makes the study of rigid body dynamics both interesting and complex.

Basic concepts

The motion of a rigid body can be described using two types of motion: translational and rotational. Translational motion refers to the movement of the center of mass of a rigid body from one point to another, while rotational motion refers to the rotation of the body around an axis.

Translational motion

Translational motion of a rigid body refers to the motion in which all the points of the body move parallel to all other points of the body and in the same direction. If the body moves in a straight line without rotating, then this motion is completely translational.

The position of a point on a rigid body in translational motion can be described using the following equation: 

r(t) = r_0 + vt
where r(t) represents the position at time t, r_0 is the initial position, and v is the velocity of the system.

Start Ending

Rotational motion

When a rigid body rotates around an axis, this motion is called rotational motion. Every point of the body moves in a circle around the axis, and the laws governing this motion are different from linear motion.

The rotational equivalent of Newton's second law is given as follows: 

τ = Iα
where τ is the torque applied to the body, I is the moment of inertia, and α is the angular acceleration.

Axis ROTATION

Equations of motion for a rigid body

The dynamics of a rigid body can be completely described using its linear momentum and angular momentum. The equations of motion governing these quantities are as follows:

  • Newton's second law for translation: 
    F = ma
  • Rotational equivalent (Newton's second law for rotation): 
    τ = Iα
  • Angular momentum: 
    L = Iω
    where L represents the angular momentum, and ω is the angular velocity.

When considering both translational and rotational motion, the total mechanical energy of a rigid body can be given as: 

E = K_translation + K_rotation E = (1/2)mv^2 + (1/2)Iω^2

Torque and moment of inertia

Torque is the rotational equivalent of force in linear motion. It is a measure of the force that tends to rotate a body about an axis. Moment of inertia is a property of a body that shows its resistance to angular acceleration when given an applied torque.

Torque:

Calculating the moment of inertia

The moment of inertia for a rigid body depends on the mass distribution relative to the axis of rotation. For simple geometries, the moment of inertia can be found using integral calculus: 

I = ∫r^2 dm
where r is the distance from the axis of a small mass element dm.

Examples and applications

Consider a thin rod rotating about one of its ends. The moment of inertia, if the mass m and length L are known, is: 

I = (1/3)mL^2
This simple case illustrates how the mass distribution affects the rotational dynamics.

Examples: gyroscope

The gyroscope is a classic example of rigid body dynamics. It consists of a rotating wheel or disk, where the axis of rotation can freely assume any direction. The stability of a gyroscope is due to the principles of angular momentum.

Suppose a gyroscope rotates at angular velocity ω. Unless an external torque is applied, the angular momentum L remains constant. This resistance to changes in angular momentum gives the gyro the ability to maintain its orientation.

spin

Real-world uses in engineering and technology

In aerospace engineering, understanding rigid body dynamics is important to the navigation and control of spacecraft, as rigid body assumptions allow for the simplicity of the complex interactions that occur during flight.

Robotics takes advantage of rigid body dynamics in designing robotic arms and structures that are capable of precise motion and task execution. Dynamics and control theory often use principles derived from this area of mechanics.

Conclusion

Rigid body dynamics is a broad topic within physics that provides fundamental insights into the behavior of dynamic objects. Its applications span myriad fields, including engineering, robotics, and astronomy. Understanding the principles governing translational and rotational motion provides powerful tools for solving complex problems in both theoretical and practical contexts.


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Rigid body mobility

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