Grade 12
  1. Advanced mechanics  1.1. Dynamics  1.1.1. Motion in two and three dimensions  1.1.2. Advanced Projectile Motion and Range Equations  1.1.3. Relative motion in different reference frames  1.1.4. Motion of charged particles in electric and magnetic fields  1.1.5. Non-uniformly accelerated motion in multiple dimensions  1.2. Dynamics  1.2.1. Newton's laws and applications in non-inertial frames  1.2.2. Dynamics of particle systems  1.2.3. Non-conservative forces and energy dissipation  1.2.4. Momentum in variable mass systems (rocket momentum)  1.2.5. Lagrangian and Hamiltonian Mechanics (Introduction)  1.3. Work, Energy and Power  1.3.1. Work done by non-conservative forces  1.3.2. Power in rotational and translational motion  1.3.3. Advanced Conservation Laws and Applications  1.3.4. Potential energy surface and stability  1.4. Advanced rotational speed  1.4.1. Torque in three dimensions  1.4.2. Moment of inertia for irregular bodies  1.4.3. Gyroscopic motion and precession  1.4.4. Rotational work–energy theorem  1.4.5. Euler's equations of rotational motion  1.5. Gravitational force  1.5.1. Advanced Orbital Mechanics  1.5.2. Lagrange points and stability of orbits  1.5.3. Gravitational time dilation  1.5.4. Escape velocity and energy considerations  1.5.5. Tidal force and the Roche limit
  2. Fluid mechanics and thermodynamics  2.1. Fluid dynamics  2.1.1. Applications of Bernoulli's Principle  2.1.2. Navier-Stokes equations and fluid resistance  2.1.3. Turbulent vs. Laminar Flow  2.1.4. Surface tension and capillary action  2.2. Advanced Thermal Physics  2.2.1. Thermodynamic equilibrium and heat engines  2.2.2. Entropy and the Second Law of Thermodynamics  2.2.3. Maxwell's demon and statistical thermodynamics  2.2.4. Thermodynamic potential and free energy
  3. Oscillations and waves  3.1. Advanced Simple Harmonic Motion  3.1.1. Coupled oscillations and common modes  3.1.2. Damped and driven oscillations  3.1.3. Phase space and resonance  3.1.4. Anharmonic oscillations  3.2. Wave motion  3.2.1. Advanced wave equations and dispersion  3.2.2. Shock Waves and Sonic Booms  3.2.3. Doppler effect in electromagnetic waves  3.2.4. Wave–particle duality
  4. Electromagnetism  4.1. Electrostatics  4.1.1. Gauss's law in unilateral charge distribution  4.1.2. Electric potential for complex charge configuration  4.1.3. Electrostatic energy and self-energy of a system  4.1.4. Dielectrics and Polarization  4.2. Current Electricity  4.2.1. Advanced Circuit Analysis (Thevenin and Norton theorems)  4.2.2. RC, RL and RLC circuits in AC and DC  4.2.3. Superconductivity and applications  4.3. Advanced Magnetism and Electromagnetic Induction  4.3.1. Magnetic fields in conductors and plasma  4.3.2. Lenz's law in time-dependent magnetic fields  4.3.3. Eddy Currents and Applications  4.3.4. Magnetic dipole moment and torque on electric current loops  4.4. Alternating Current and Electromagnetic Waves  4.4.1. LC Circuits and Resonance  4.4.2. Maxwell's equations and electromagnetic wave transmission  4.4.3. Applications of Electromagnetic Waves  4.4.4. Polarization and reflection of EM waves
  5. Optics  5.1. Geometrical Optics  5.1.1. Aberrations in optical systems  5.1.2. Advanced lens and mirror formulas  5.1.3. Optical Instruments and Adaptive Optics  5.2. Wave optics  5.2.1. Diffraction at single and multiple slits  5.2.2. Optical Coherence and Interferometry  5.2.3. Holography and applications
  6. Modern and quantum physics  6.1. Special relativity  6.1.1. Time expansion and length contraction  6.1.2. Relativistic energy and mass  6.1.3. Relativistic Doppler effect  6.2. Quantum mechanics  6.2.1. Schrödinger equation and wave function  6.2.2. Quantum tunneling and applications  6.2.3. Heisenberg uncertainty principle  6.2.4. Quantum superposition and entanglement  6.3. Nuclear physics  6.3.1. Binding energy and mass defect  6.3.2. Neutrino physics and beta decay  6.3.3. Nuclear Fission and Fusion  6.4. Particle physics  6.4.1. Standard Model and fundamental interactions  6.4.2. The Higgs boson and symmetry breaking  6.4.3. Antimatter and CP violation
  7. Astrophysics and cosmology  7.1. General relativity and gravity  7.1.1. Einstein's field equations  7.1.2. Gravitational waves and their detection  7.1.3. Black holes and event horizons  7.2. Cosmology  7.2.1. Dark matter and dark energy  7.2.2. Cosmic microwave background radiation  7.2.3. The expansion of the universe and Hubble's law

Grade 12 ↓


Advanced mechanics


Advanced mechanics is a fascinating and vast subject in physics, dealing with the motion and behavior of objects. In this section, we will explore the fundamentals and concepts of advanced mechanics, focusing mainly on phenomena that go beyond basic principles such as Newton's laws of motion.

Newtonian mechanics recap

Mechanics begins with Newton's three laws, which provide the basis for understanding motion. In short:

  • Newton's First Law: An object will remain at rest or in uniform motion unless an external force is applied to it.
  • Newton's second law: The force acting on an object is equal to the product of the object's mass and its acceleration. This is expressed in the formula F = ma.
  • Newton's Third Law: For every action there is an equal and opposite reaction.

Although these laws are fundamental, advanced mechanics deepens our understanding by considering more complex scenarios such as nonlinear systems, rotating frames of reference, and rigid body dynamics.

Rotational motion

Unlike linear motion, rotational motion involves the movement of objects in circular or paths that involve a change in angular direction. Some important parameters are:

  • Angular displacement: The angle through which an object has turned or moved.
  • Angular velocity: The rate of change of angular displacement with respect to time, usually denoted by ω.
  • Angular acceleration: The rate of change of angular velocity.

Formulas for rotational motion are often similar to formulas for linear motion, but they are based on angles rather than distances:


θ = ω₀t + 0.5αt²
ω = ω₀ + αt
ω² = ω₀² + 2αθ

Example of rotational motion

Imagine a rotating wheel. If the wheel is initially stationary and it accelerates at a rate of 5 radians/second² for 10 seconds, we can calculate the final angular velocity using:

ω = ω₀ + αt
ω = 0 + (5 rad/s²)(10 s) = 50 rad/s

This means that after 10 seconds the wheel starts rotating at a speed of 50 radians per second.

ω = 50 radian/second

Moment of inertia

The moment of inertia, denoted by I, plays a role in rotational dynamics similar to the role mass plays in linear motion. It is a measure of an object's resistance to a change in rotation and depends on the mass distribution relative to the axis of rotation.

For solid objects, the moment of inertia can be calculated using integration over the volume of the object. For simple shapes, standard formulas exist:

  • Solid sphere: I = (2/5)MR²
  • Hollow cylinder: I = MR²
  • Solid disk: I = (1/2)MR²

Torque

Torque is the rotational analog of force, and measures how much a force applied to an object rotates that object. It is given as:

τ = rFsinθ

where τ is the torque, r is the distance from the pivot point to the point where the force is applied, F is the magnitude of the force, and θ is the angle between the force vector and the lever arm.

Consider using a wrench to turn a bolt. The torque applied when applying force near the end of the wrench is greater than when applying force near the head of the bolt.

Force R

Conservation of angular momentum

Angular momentum, denoted by L, is conserved in an isolated system, just as linear momentum is conserved. If no external torque acts on a system, the total angular momentum remains constant:

L = Iω

Imagine an ice skater spinning. As they pull their arms in, their moment of inertia decreases, and they spin faster to conserve angular momentum. Their angular momentum is the same before and after they pull their arms in.

Gravity and orbits

Gravitation governs the motion of celestial bodies and is an important component of advanced mechanics. The universal law of gravitation is represented by:

F = G(m₁m₂/r²)

where F is the gravitational force between the two objects, G is the gravitational constant, m₁ and m₂ are their masses, and r is the distance between their centers. Orbits can be predicted by considering the gravitational effect, which leads to elliptical paths described by Kepler's laws.

Energy in advanced mechanics

Energy conservation in mechanics extends to potential and kinetic energy in rotational processes, which is expressed as:

KE = 0.5Iω²
PE = mgh

These forms of energy are converted into each other but are conserved within an isolated system.

Example calculation: energy conservation

A solid disc of mass 10 kg and radius 0.5 m is rotating at a speed of 10 rad/s. Calculate its kinetic energy.

I = (1/2)MR² = (1/2)(10 kg)(0.5 m)²
I = 1.25 kg·m²
KE = 0.5Iω² = 0.5(1.25 kg·m²)(10 rad/s)²
KE = 62.5 J

Therefore, the kinetic energy of the rotating disc is 62.5 J.

Advanced motion in fluids

Understanding motion in fluids involves concepts such as viscosity and laminar versus turbulent flow. The Navier-Stokes equations describe how velocity fields evolve in a fluid with specified boundary conditions:

ρ(∂v/∂t + v·∇v) = -∇p + ∇·T + f

where ρ is the density of the fluid, v is the velocity vector, p is the pressure, T denotes the shear stress, and f denotes the external forces.

Advanced problems in mechanics

Mechanics can be extended to more complex applications, such as the analysis of chaotic systems, where predictions are difficult even though they are governed by deterministic rules, and many-body problems, where many forces interact.

Conclusion

Advanced mechanics explores concepts far beyond elementary physics, including the important forces and motions in our universe. Understanding these complex ideas expands our understanding of the physical world, which is important for those interested in physics and engineering.


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