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 motion is a fundamental concept in classical mechanics that describes the motion of objects or systems that rotate around a central axis. While linear motion deals with objects moving along a path in a straight line, rotational motion involves objects that spin or revolve.

Understanding rotational motion is important for analyzing systems ranging from spinning tops and gears to the rotation of planets and stars. Several key concepts and equations describe rotational motion, which are similar to the equations describing linear motion, but with different variables.

Angular displacement, velocity and acceleration

Similar to linear displacement, velocity, and acceleration, rotational motion involves angular displacement, angular velocity, and angular acceleration.

Angular displacement is the angle through which a point or line has been rotated in a specified sense about a specified axis. It is measured in radians.

Δθ = θ_f - θ_i

where Δθ is the angular displacement, θ_f is the final angular position, and θ_i is the initial angular position.

Angular velocity (ω) is the rate at which an object rotates. It is described by the change in angular displacement over time and is measured in radians per second (rad/s).

ω = Δθ / Δt

where ω is the angular velocity, Δθ is the angular displacement, and Δt is the time interval.

Angular acceleration (α) tells how quickly the angular velocity changes with time. It is measured in radians per second squared (rad/s²).

α = Δω / Δt

where α is the angular acceleration, Δω is the change in angular velocity, and Δt is the time interval.

Visualization of rotational motion

Imagine a rotating wheel. The wheel has a central axis and rotates around it. To understand these concepts further, consider the following illustration:

Δθ

In this representation, the black lines represent the reference and current angular positions of a point on the wheel, and the red line represents the angular displacement, Δθ.

Equations of rotational motion

The equations describing rotational motion have the same form as the equations of linear motion. However, they use an angular analog.

  • Final angular velocity: ω_f = ω_i + αt
  • Angular displacement: θ = ω_i t + 0.5αt^2
  • Square of final angular velocity: ω_f^2 = ω_i^2 + 2αθ

In these formulas, ω_i is the initial angular velocity, ω_f is the final angular velocity, α is the angular acceleration, and θ is the angular displacement.

Moment of inertia

Moment of inertia, denoted as I, is a measure of an object's resistance to a change in its rotational speed. It depends on the mass distribution of the object relative to the axis of rotation.

The simple equation for a point mass is:

I = mr²

Where m is the mass of the object and r is the distance of the mass from the axis of rotation.

For more complex objects, the moment of inertia requires summing or integration of all mass elements:

I = Σ(m_i r_i²)

or, for a persistent object:

I = ∫ r² dm

Torque

Torque is the rotational analog of force. It measures how much a force applied to an object rotates it.

τ = r × F

Where τ is the torque, r is the position vector from the axis of rotation to the point where the force is applied, and F is the force vector. The cross product indicates that the torque depends on both the magnitude and direction of the force, as well as the distance from the axis.

Newton's second law for rotational motion

As with linear motion, a form of Newton's second law applies to rotational motion. It states that the net torque acting on an object is equal to the product of its moment of inertia and angular acceleration.

Στ = Iα

Example problems

Let us consider some practical examples of solving rotational motion problems to strengthen our understanding.

Example 1: Spinning disc

A disc of radius 0.5 m and mass 2 kg is initially at rest. A force of 10 N is applied tangentially to its edge. Find the angular acceleration.

First, calculate the moment of inertia of the disc using the formula for a solid disc:

I = 0.5 * m * r²

I = 0.5 * 2 kg * (0.5 m)² = 0.25 kg·m²

Apply the torque formula:

τ = r × F = 0.5 m × 10 N = 5 N·m

Use Newton's second law for rotation:

Στ = Iα => 5 N·m = 0.25 kg·m² * α

α = 5 N·m / 0.25 kg·m² = 20 rad/s²

Example 2: Rotating rod

A thin rod of length 1 m and mass 3 kg is fixed at one end and is free to rotate. A force of 15 N is applied perpendicularly at the centre of the rod. Calculate the angular acceleration.

Calculate the moment of inertia about the pivot at one end:

I = (1/3) * m * L²

I = (1/3) * 3 kg * (1 m)² = 1 kg·m²

Calculate the torque:

τ = r × F = 0.5 m × 15 N = 7.5 N·m

Apply Newton's second law:

Στ = Iα => 7.5 N·m = 1 kg·m² * α

α = 7.5 rad/s²

Conclusion

Rotational motion is important in understanding many physical phenomena and engineering systems. By mastering key concepts such as angular velocity, angular acceleration, torque, and moment of inertia, you can analyze and predict complex rotational behaviors.

Continued exploration of rotational motion will not only enrich the understanding of classical mechanics principles but will also lead to advances in practical applications such as machinery design and astrophysics.


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Rotational motion

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