Grade 11
  1. Mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two and three dimensions  1.1.3. Displacement and Distance  1.1.4. Speed and Velocity  1.1.5. Acceleration and deceleration  1.1.6. Graphical analysis of motion  1.1.7. Equations of motion (advanced derivations)  1.1.8. Free fall and terminal velocity  1.1.9. Projectile motion  1.1.10. Relative velocity in one and two dimensions  1.1.11. Motion of a car on an inclined plane  1.2. Dynamics  1.2.1. Newton's laws of motion and their applications  1.2.2. Inertia and Newton's first law  1.2.3. Force and types of force  1.2.4. Static and kinetic friction  1.2.5. Circular motion and centripetal acceleration  1.2.6. Non-uniform circular motion  1.2.7. Banking of roads and curves  1.2.8. Momentum and Impulse  1.2.9. Conservation of momentum in one and two dimensions  1.2.10. Applications of Newton's Third Law  1.3. Work, Energy and Power  1.3.1. Work done by a constant and variable force  1.3.2. Work–energy theorem  1.3.3. Kinetic and potential energy  1.3.4. Gravitational and elastic potential energy  1.3.5. Conservation of mechanical energy  1.3.6. Power and instantaneous power  1.3.7. Efficiency of machines  1.3.8. Work done by conservative and non-conservative forces  1.4. Rotational motion  1.4.1. Torque and angular acceleration  1.4.2. Rotational equilibrium  1.4.3. Moment of inertia and its applications  1.4.4. Angular momentum and its conservation  1.4.5. Kinetic energy of rolling motion and rotation  1.4.6. Parallel and Perpendicular Axis Theorem  1.4.7. Dynamics of rotational motion
  2. Gravitational force  2.1. Universal gravitation  2.1.1. Newton's law of universal gravitation  2.1.2. Gravitational field and potential  2.1.3. Change in acceleration due to gravity  2.1.4. Kepler's laws of planetary motion  2.1.5. Orbital mechanics and satellites  2.1.6. Escape velocity and orbital velocity  2.1.7. Weightlessness and free fall  2.1.8. Gravitational potential energy and energy in orbits  2.1.9. Black holes and gravitational lensing
  3. Properties of matter  3.1. Fluid mechanics  3.1.1. Density and pressure in liquids  3.1.2. Pascal's theory and applications  3.1.3. Archimedes' Principle and Buoyancy  3.1.4. Bernoulli's equation and applications  3.1.5. Continuity equation and the Venturi effect  3.1.6. Surface tension and capillarity  3.1.7. Viscosity and Poiseuille's Law  3.1.8. Reynolds number and turbulence  3.2. Elasticity and deformation  3.2.1. Hooke's law and the stress-strain relation  3.2.2. Elastic modulus and applications  3.2.3. Deformation and yield strength of solids
  4. Thermal physics  4.1. Heat and temperature  4.1.1. Thermometric properties and temperature scale  4.1.2. Thermal expansion of solids, liquids and gases  4.1.3. Heat transfer system  4.1.4. Specific heat capacity and calorimetry  4.1.5. Latent heat and phase change  4.2. Kinetic theory of gases  4.2.1. Molecular model of gases  4.2.2. Kinetic explanation of temperature  4.2.3. Ideal Gas Equation and Deviations  4.2.4. Mean free path and Maxwell–Boltzmann distribution  4.3. Laws of Thermodynamics  4.3.1. Zeroth Law and Thermal Equilibrium  4.3.2. First Law and Internal Energy  4.3.3. The Second Law and Entropy  4.3.4. Carnot cycle and heat engines  4.3.5. Refrigerators and heat pumps
  5. Waves and oscillations  5.1. Simple Harmonic Motion  5.1.1. Equations of SHM  5.1.2. Energy in SHM  5.1.3. Damped and forced oscillations  5.1.4. Resonance and applications  5.1.5. Pendulum and spring-mass system  5.2. Wave motion  5.2.1. Types of mechanical waves  5.2.2. Wave motion and energy transport  5.2.3. Superposition Principle and Standing Waves  5.2.4. Reflection, Refraction and Diffraction  5.2.5. Doppler effect in sound and light  5.2.6. Pulsations and interference
  6. Electricity and Magnetism  6.1. Electrostatics  6.1.1. Electric charge and conservation  6.1.2. Coulomb's law and its applications  6.1.3. Electric field and electric flux  6.1.4. Electric potential and potential energy  6.1.5. Capacitance and Energy Stored in a Capacitor  6.2. Current Electricity  6.2.1. Ohm's law and electrical resistance  6.2.2. Resistivity and temperature dependence  6.2.3. Kirchhoff's Laws and Circuit Analysis  6.2.4. Electrical power and efficiency  6.2.5. Combination of resistors  6.3. Magnetism and Electromagnetism  6.3.1. Magnetic fields and their sources  6.3.2. Magnetic force on moving charges  6.3.3. Electromagnetic induction  6.3.4. Faraday's and Lenz's laws  6.3.5. Inductance and Transformer  6.3.6. Alternating current and its applications
  7. Optics  7.1. Reflection and Refraction  7.1.1. laws of reflection and refraction  7.1.2. Mirrors and lenses  7.1.3. Total internal reflection and optical fiber  7.2. Wave optics  7.2.1. Interference and Diffraction  7.2.2. Young's double-slit experiment  7.2.3. Polarization of light
  8. Modern Physics  8.1.1. Photoelectric effect and Einstein's theory  8.1.2. Wave–particle duality  8.1.3. Heisenberg's uncertainty principle  8.1.4. Compton Effect  8.2. Atomic and Nuclear Physics  8.2.1. Atomic model and energy levels  8.2.2. Radioactive decay and half-life  8.2.3. Nuclear Fission and Fusion
  9. Electronics and Communication  9.1. Semiconductors  9.1.1. pn junction and diode  9.1.2. Transistors and Logic Gates  9.1.3. Integrated Circuits and Applications  9.2. Communication Systems  9.2.1. Basics of Analog and Digital Communication  9.2.2. Wireless and Optical Communication  9.2.3. Modulation and Demodulation

Grade 11 → Mechanics → Rotational motion ↓


Angular momentum and its conservation


Angular momentum is an important concept in physics, especially when dealing with rotational motion. In the world around us, many objects rotate: the Earth spins on its axis, the wheels of a car spin, and figure skaters spin on the ice. Angular momentum helps us describe and predict the motion of these rotating objects.

Understanding angular momentum

In simple terms, angular momentum is the rotational equivalent of linear momentum. While linear momentum deals with objects moving in straight lines, angular momentum deals with objects that are rotating or moving.

The angular momentum L for an object rotating about an axis can be described by the following formula:

L = I ω

Where:

  • L is the angular momentum.
  • I is the moment of inertia.
  • ω (omega) is the angular velocity.

Let us take a closer look at these factors:

Moment of inertia

The moment of inertia I is a measure of how difficult it is to change the rotational speed of an object. It depends on the mass of the object and the distribution of that mass relative to the axis of rotation. The formula for the moment of inertia varies depending on the shape and mass distribution. For a particle, it is:

I = mr^2

For an extended body, this can get a little more complicated, involving integrals, but the main point is that larger or more extended bodies have greater moments of inertia.

Angular velocity

Angular velocity ω is a measure of the speed of rotation of an object. It describes the instantaneous speed of rotation and is usually measured in radians per second.

Law of conservation of angular momentum

According to the law of conservation of angular momentum, if no external torque acts on a system, then the total angular momentum of the system remains constant.

This is similar to conservation of linear momentum, which says that the total linear momentum of a closed system remains constant unless an external force acts on it.

Formula representation

Mathematically, conservation of angular momentum can be expressed as:

L_initial = L_final

This means that if there is no external torque then the initial angular momentum of the system will be equal to the final angular momentum.

Visual example

Spinning wheel

Example: Think of a bicycle wheel rotating on its axis. The rotation of the wheel gives it angular momentum. If you stop applying external force, it will continue rotating due to conservation of angular momentum.

Ice skater

Example: When a figure skater pulls their arms in, their moment of inertia decreases. According to conservation of angular momentum, as I decreases, ω must increase, causing them to spin faster.

Examples of conservation of angular momentum

Motion of planets

Consider the planets orbiting the Sun. Each planet has angular momentum with respect to its orbit around the Sun.

If no external torque acts on the planet-Sun system, the angular momentum of the planet remains constant. This is why planets remain in stable orbits and why their speed and distance from the Sun are inversely related; as a planet moves closer to the Sun, its speed increases and as it moves away, its speed slows down, so that the total angular momentum remains constant.

Neutron stars

Neutron stars, which are the remnants of massive stars, dramatically demonstrate the conservation of angular momentum. When a star collapses under its own gravity to become a neutron star, its radius decreases rapidly, reducing its moment of inertia.

Because its mass is concentrated in a very small volume, a neutron star rotates extremely rapidly with a high angular velocity to conserve angular momentum.

Trampoline rotation

Imagine you are flipping forward on a trampoline. While you are moving through the air, you are not touching anything that could apply an external torque. Assuming there is no air resistance, your angular momentum is constant.

If you start in a bent position, with your arms and legs pulled inward, you will rotate faster than if you extended your limbs outward, spreading your mass away from the axis of rotation. This is because pulling your limbs inward decreases the moment of inertia, so angular velocity must increase for angular momentum conservation.

Conclusion

Understanding angular momentum and its conservation provides a powerful insight into the workings of our universe. From the spin of a figure skater to the orbits of celestial bodies, the principles of angular momentum guide the path of rotating systems.

By understanding these concepts, we can better predict and analyze the behavior of objects in rotation, shedding light on the complex dance of matter in the universe and near us.


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Angular momentum and its conservation

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