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 ↓


Torque and Angular Momentum


In classical mechanics, it is important to understand the concepts of torque and angular momentum when studying rotational motion. They are the rotational analogues of force and momentum in linear motion and provide insight into how objects behave when they rotate. In this lesson, we will explore these concepts in detail using simple explanations, equations, and examples. The goal is to provide a comprehensive understanding for undergraduate physics students.

What is torque?

Torque, also called moment of force, is a measure of the tendency of a force to rotate an object about an axis. Just as force is needed to change the linear velocity of an object, torque is needed to change the angular velocity of an object. The magnitude of torque depends on three factors: the magnitude of the force, the distance from the pivot point (lever arm or moment arm), and the angle at which the force is applied.

τ = r × F = rFsin(θ)

In this equation:

  • τ is the torque.
  • r is the distance from the axis of rotation to the point where the force is applied.
  • F is the magnitude of the force.
  • θ is the angle between the position vector and the direction of the force.

Visual Example: Torque

The concept of torque can be understood by considering a door. To open the door, you apply a force some distance away from the hinge. The torque you apply is greatest when you push perpendicular to the surface of the door, causing θ to be equal to 90 degrees, which makes sin(θ) maximum.

Door Force(F)

Calculating torque in practice

To understand the practicality of torque, let's look at another example. Consider a wrench that is turning a nut. The force applied to the wrench is directed along its length, and the farther away you hold the wrench, the greater the torque as the perpendicular distance r increases. If you apply a force of 10 N at a distance of 0.3 m from the pivot, the torque is:

τ = rF = 0.3 m × 10 N = 3 N·m

What is angular momentum?

Angular momentum is a vector quantity that represents the amount of rotation of an object and is an essential fundamental quantity in physics. For a point mass, it is defined as the cross-product of the position vector and the speed. In many ways, angular momentum is similar to linear momentum. Just as linear momentum measures the amount of linear motion, angular momentum measures the amount of rotational motion.

L = r × p = r × mv

In this equation:

  • L is the angular momentum.
  • r is the position vector from the axis of rotation to the point mass.
  • p is the linear momentum of the point mass.
  • m is the mass of the object.
  • v is the linear velocity of the object.

Visual Example: Angular Momentum

Imagine a rotating wheel. The mass of the wheel is distributed at a fixed radius from the axis; as it rotates due to some initial force or torque, it acquires angular momentum. If no external torque acts on it, its angular momentum remains constant.

Velocity (v) radius (r)

Conservation of angular momentum

If no external torque acts on a system then its angular momentum remains constant. This is known as conservation of angular momentum. This principle explains many phenomena in daily life and nature. For example, when a figure skater pulls her arms in, they rotate faster. As the arms retract, their moment of inertia decreases while their angular velocity increases.

Example of Conservation

Consider a skater who is spinning with his arms outstretched. Let I_i be the initial moment of inertia and ω_i be the initial angular velocity. By conservation of angular momentum:

L_i = L_f I_i ω_i = I_f ω_f

where I_f and ω_f are the final moment of inertia and angular velocity when the arms are pulled back.

Relation between torque and angular momentum

Torque and angular momentum are closely related. The rate of change of the angular momentum of a body is equal to the net external torque applied to it. Mathematically it is expressed as:

τ = dL/dt

This equation shows that if you know the angular momentum of a system as a function of time, you can determine the torque acting on it, and vice versa.

Example problem: disk rotation

Let us solve a practical problem. Consider a disc rotating about its center. Its mass is 2 kg and its radius is 0.5 m. A force of 4 N is applied tangentially to its edge. We want to determine the angular momentum and torque of the disc.

  • Step 1: Calculate the torque:
τ = rF = 0.5 m × 4 N = 2 N·m
  • Step 2: Calculate the angular acceleration α using τ = Iα, where I = 1/2 mr^2 for a disk:
I = 1/2 × 2 kg × (0.5 m)^2 = 0.25 kg·m² α = τ/I = 2 N·m / 0.25 kg·m² = 8 rad/s²
  • Step 3: Calculate the angular momentum at t = 1 s. Assuming the initial angular velocity ω_0 is zero:
ω = ω_0 + αt = 0 + 8 × 1 = 8 rad/s L = Iω = 0.25 kg·m² × 8 rad/s = 2 kg·m²/s

Final thoughts

Torque and angular momentum are important within rotational motion in classical mechanics. They allow us to describe how rotating objects move, and understanding their interaction is essential for solving problems in physics ranging from simple mechanical systems to celestial mechanics. By understanding these concepts, students can predict and analyze the behavior of rotating bodies in a variety of contexts.

In short, torque provides the ability to initiate a change in rotational motion, while angular momentum expresses a state of motion that is maintained unless acted upon by an external torque. Their understanding broadens the scope of analyzing physical systems dynamically and is foundational knowledge in the study of mechanics.


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