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 → Rotational motion ↓


Rotational Kinematics


Rotational dynamics is a branch of physics that describes the motion of points, bodies, and systems of bodies without considering the forces that cause them to move. It is the rotational analogue of linear dynamics. In this field, we deal mainly with angular displacement, angular velocity, and angular acceleration. These concepts are essential for understanding everything from the rotation of a planet to the rotation of a wheel.

Basic concepts

Angular displacement

In linear motion, displacement refers to a change in position. Similarly, in rotational motion, angular displacement refers to the angle through which an object rotates around a particular axis. It is usually represented by the symbol θ (theta).

Angular displacement is measured in radians, although degrees are sometimes used. One full rotation is equal to 2π radians or 360 degrees.

Example:

Consider a turntable rotating on its axis. If it starts at zero degrees and rotates 90 degrees, the angular displacement is:

 θ = 90° = π/2 radians

Visual example:

θ

Angular velocity

Angular velocity represents the rate of change of angular displacement. It is a vector quantity, having both magnitude and direction, although in basic cases we consider only its magnitude. It is represented by ω (omega).

 ω = θ / T

Where θ is the angular displacement and t is the time taken.

Example:

If a wheel rotates π radians in 2 seconds, then the angular velocity is:

 ω = π / 2 radian/second

Visual example:

ω

Angular acceleration

Angular acceleration is the rate of change of angular velocity relative to time. Similar to linear acceleration, it tells us how quickly an object is speeding up or slowing down its rotation. It is often represented by α (alpha).

 α = Δω / Δt

Where Δω is the change in angular velocity and Δt is the change in time.

Example:

If the angular velocity of a spinning top increases from 0 rad/s to 2π rad/s in 4 sec, then the angular acceleration is:

 α = (2π - 0) / 4 = π/2 radian/second²

Visual example:

α

Equations of rotational motion

Just as there are equations of motion for linear systems, there are parallel equations for rotating systems. These equations relate angular displacement, angular velocity, angular acceleration, and time.

First equation of motion

This equation is useful when angular displacement needs to be determined but the acceleration is not zero:

 ω = ω₀ + αt

where ω₀ is the initial angular velocity.

Second equation of motion

To find the angular displacement we use this equation:

 θ = ω₀t + 0.5*αt²

Third equation of motion

This equation allows us to find the final angular velocity without considering time:

 ω² = ω₀² + 2αθ

Applications of rotational dynamics

Example problems

Problem 1: A wheel starts from rest and accelerates with angular acceleration

 α = 2 radians/second²

What is its angular velocity after 3 seconds?

Solution:

Using the first equation of motion:

 ω = ω₀ + αt

given

 ω₀ = 0 rad/s, α = 2 rad/s², t = 3 s

Thus,

 ω = 0 + 2 * 3 = 6 radian/second

Problem 2: A fan is turned off, and it takes 10 seconds to stop. If the initial angular velocity of the fan is

 25 rad/s

Find the angular acceleration.

Solution:

 ω = ω₀ + αt

Here,

 0 = 25 + α * 10

Solve for α:

 α = -25 / 10 = -2.5 radians/second²

Adding linear and rotational quantities

It's important to know how rotational and linear quantities are related, especially when working with real-world applications such as wheels and gears.

Linear displacement s and angular displacement θ :

 s = rθ

where r is the radius of the circle.

Linear velocity v and angular velocity ω :

 v = rω

Linear acceleration a and angular acceleration α :

 a = rα

Example:

A bicycle wheel with a radius of 0.5 m rotates forward at an angular velocity of 4 radians/second. What is the linear velocity?

 v = rω = 0.5 * 4 = 2 m/s

Summary and conclusion

Understanding rotational dynamics is important when studying systems involving rotation. Whether it's planets orbiting stars, wheels turning on vehicles, or the spinning of atoms, the principles of angular displacement, velocity, and acceleration remain consistent. By mastering these fundamental concepts, you can analyze and predict the motion of rotating objects in many fields within physics and engineering, making them invaluable in both academic and practical applications.


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Rotational Kinematics

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