Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Classical mechanics → Newton's Laws of Motion ↓


Constraints and pseudo-forces


Introduction

It is important to understand the concepts of constraints and pseudo-forces when studying Newton's laws of motion in classical mechanics. These concepts help solve problems involving objects that are either restricted in some way or that are analyzed from non-inertial reference frames. This explanation delves deep into the nature of constraints and pseudo-forces, which is illustrated with textual examples and visual representations to promote clear and comprehensive understanding.

Constraints in mechanics

In classical mechanics, a constraint is a condition that restricts the motion of a particle or system of particles. Constraints are necessary because they represent physical limits imposed by the environment or interactions in the system.

Common types of constraints include unilateral and bilateral constraints. Unilateral constraints restrict motion in one direction (for example, a ball on a flat surface that cannot penetrate the surface), while bilateral constraints restrict motion in two or more directions (such as a bead sliding on a wire or rod).

In addition, restrictions can be holonomic or non-holonomic. Holonomic restrictions are those that can be expressed as explicit functions of coordinates and time. For example, a pendulum with a fixed-length string has a restriction given by:

L = constant

On the other hand, non-holonomic restrictions involve inequalities or differential conditions, such as the no-slip condition of a rotating wheel.

Visual example of constraints

taut wire surface

In the visualization above, a red ball is attached to a blue line that represents a string constraint. The ball can swing back and forth but cannot move vertically due to the tension in the string. Additionally, it cannot pass through a black line that represents a solid surface. This shows a combination of bilateral and holonomic constraints.

Mathematical representation of constraints

To express constraints mathematically, consider a system with coordinates (x_1, x_2, ldots, x_n). A holonomic constraint can be formulated as a function:

f(x_1, x_2, ..., x_n, t) = 0

For example, if a particle is to stay on a circle of radius R centered at the origin, the restriction is:

x^2 + y^2 - R^2 = 0

Dealing with constraints often requires the use of Lagrange multipliers in mechanics, so that these conditions can be effectively incorporated into the equations of motion.

Pseudo forces

Pseudo forces, also called fictitious forces, arise when analyzing motion from a non-inertial (accelerating) reference frame. These forces are not real, but are introduced to take into account the acceleration of the frame.

A classic example of a pseudo force is the centrifugal force that appears when analyzing circular motion from a rotating reference frame. If you are inside a car that is taking a sharp turn, you may feel a push from the side of the car, which is actually the centrifugal force acting away from the center of the car's turning path.

Visual example of pseudo forces

centrifugal force Center

In this illustration, a point on the circumference of a circle represents an object in a rotating frame. To an observer rotating with the object, a centrifugal pseudo-force appears to act outward, even though there is no such force in reality. This is only an assumption because the frame of reference itself is non-inertial.

Analysis of pseudo powers

To measure the pseudo force, it is necessary to consider the acceleration a of the non-inertial reference frame. If the mass m is in this frame, then the pseudo force F_p applied to it is:

F_p = -m * a

The negative sign indicates that the pseudo force is always in the opposite direction of the accelerating frame. For example, when an elevator starts to descend, you feel lighter because of the pseudo force acting upward.

Conclusion

Constraints and pseudo forces are fundamental elements in the study of dynamics. While constraints determine the permissible motions of a system, pseudo forces help in understanding mechanics from non-inertial frames. A thorough understanding of these concepts is crucial for solving complex problems in classical mechanics that involve systems affected by various forces and motion restrictions.

Through continued study and practice, theories related to constraints and pseudo forces become an integral part of the analysis of physical systems in a variety of contexts, enhancing the ability to understand, predict, and explain the behavior of objects under a variety of conditions.


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Constraints and pseudo-forces

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