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 ↓


Moment of inertia


The concept of moment of inertia is a fundamental aspect of rotational motion in classical mechanics, just as mass is for linear motion. It is a measure of an object's resistance to changes in its rotation. To better understand moment of inertia, let's look at its definition, mathematical formulation, physical significance, and examples that make it easier to understand.

Conceptual understanding

Imagine that you are trying to rotate two different wheels. One is a bicycle wheel and the other is a solid wooden wheel. Intuitively, you will feel that the bicycle wheel is easier to rotate than the heavy wooden wheel. This difference in effort is due to the moment of inertia, which is associated with both the mass of the object and its distribution relative to the axis of rotation.

Mathematical definition

The moment of inertia is defined mathematically as:

I = Σ m i r i 2

Here:

  • I is the moment of inertia.
  • m i is the mass of each particle in the object.
  • r i is the distance of each mass from the axis of rotation.

Visual example

R

In this example, the black circle at the center represents the axis of rotation and the red circle is a point mass located on the wheel. The distance r from the axis determines its contribution to the moment of inertia.

Physical interpretation

Moment of inertia is effectively the rotational analogy of mass in linear motion. Just as a larger mass requires more force to accelerate in a straight line, a larger moment of inertia requires more torque to achieve angular acceleration.

Factors affecting the moment of inertia

There are two main factors behind this:

  1. The mass of the object. Heavier objects have greater moment of inertia.
  2. The distribution of that mass relative to the axis of rotation. Mass distributed away from the axis significantly increases the moment of inertia.

Comparison example

Consider a rod rotating on an axis perpendicular to its length:

If the axis is at the centre, I is small because the mass is uniformly distributed. If it rotates about one end, I of the same rod will be large because more mass is away from the axis.

Moment of inertia formulas for common shapes

The moment of inertia for different shapes can be calculated using different formulas.

Point mass

I = mr 2

Solid cylinder or disc with rotation axis through the center

I = (1/2) MR 2

Solid sphere with axis of rotation through the centre

I = (2/5) MR 2

Rod with axis of rotation passing through the centre, perpendicular to the length

I = (1/12) ML 2

Importance of moment of inertia in real life applications

The concept of moment of inertia is important in many real-world applications:

  • Engineering and Design: Understanding moment of inertia in the design of machinery and vehicles helps optimize performance, stability, and energy efficiency.
  • Sports: Athletes in sports such as diving or gymnastics use moment of inertia to control their turns and flips by adjusting their body position.
  • Astronomy: The rotation of planets and stars is governed by the principles of moment of inertia.
  • Manufacturing: Industrial machines depend on the proper application of torque and moment of inertia to function effectively.

Conclusion

Moment of inertia is a widespread measure of rotational dynamics. Understanding it is essential for the study and application of physics in various fields, ensuring proper mechanical functioning, and analyzing motions. By breaking it down into its fundamentals and formulas, we can appreciate both its theoretical aspects and its practical importance.


Undergraduate → 1.5.2


U
username
0%
completed in Undergraduate

← Prev (1.5.1)
Torque and Angular Momentum
Next (1.5.3) →
Rotational Kinematics

Comments 



Moment of inertia

https://www.physicsflow.com/ug/1.5.2?pfal=en