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 → Waves and oscillations → Simple Harmonic Motion ↓


Damped and forced oscillations


Introduction

Oscillations are an integral part of our daily lives and nature. From the way a clock pendulum swings to the vibration of our vocal cords while speaking. Understanding oscillations can provide us with information about various natural phenomena and technological applications. Simple Harmonic Motion (SHM) has two types of oscillations, damped oscillations and forced oscillations. In this lesson, we will explain these concepts in a simple and easy to understand manner.

Understanding Simple Harmonic Motion (SHM)

Before diving into damped and forced oscillations, let us refresh our understanding about simple harmonic motion (SHM). SHM is a type of oscillatory motion where the restoring force is directly proportional to the displacement from the mean position and acts in the opposite direction to the displacement. It can be mathematically expressed as:

F = -kx

Here, F is the restoring force, k is the spring constant, and x is the displacement from the mean position. The negative sign indicates that the force is in the opposite direction of the displacement.

Let's look at an example of a simple harmonic oscillator:


    
    
    
    Average position
    ↔ F
    ↔ X

In the visualization, the blue circle represents an oscillating object that moves back and forth about a mean position. The arrows represent vector quantities: displacement x in green and force F in red.

Damped oscillation

In the ideal world, the oscillations would continue forever. However, in real life, oscillating systems are subject to forces such as friction and air resistance, which cause them to lose energy over time. This type of motion is called damped oscillation.

Types of damping

There are different degrees of damping depending on the medium and forces:

  • Hypodamped: The system oscillates with a gradually decreasing amplitude.
  • Critically damped: The system returns to the equilibrium position quickly without oscillation.
  • Overdamped: The system returns to equilibrium slowly, without oscillation.

The mathematical expression for damped oscillations is:

x(t) = A e^(-bt/2m) cos(ωt + φ)

Here, A is the initial amplitude, b is the damping coefficient, m is the mass, t is the time, ω is the angular frequency, and φ is the phase angle.

Let us imagine a slightly damped oscillation:


    
    
    
    Time
    Dimensions

In the visualization above, the red curve shows how the amplitude decreases with time in an underdamped system.

Practical example of damped oscillations

Consider the car's suspension system. The springs and shock absorbers in the suspension are designed to reduce vibrations caused by bumps and potholes in the road, making the ride more comfortable.

Forced oscillation

In forced oscillation, an external force continuously acts on the system, giving it energy. The system oscillates under the influence of an external periodic force.

F(t) = F_0 cos(ω_d t)

Here, F(t) is the external periodic force, F_0 is the amplitude of the force, and ω_d is the driving angular frequency.

Resonance in forced oscillation

Resonance is an important concept related to forced oscillation, which occurs when the frequency of the external force matches the natural frequency of the system. This can cause a significant increase in amplitude, which can sometimes lead to structural failures such as bridges collapsing due to marching soldiers or when a singer breaks a glass by singing at its natural frequency.

Practical example of forced oscillation

A simple example of forced oscillation is pushing a swing at regular intervals. The external force applied by the person pushing keeps the swing moving.

Let us imagine the resonance phenomenon:


    
    
    
    Time
    Dimensions

The blue curve shows the increase in amplitude due to resonance.

Conclusion

Understanding damped and forced oscillations opens the door to exploring complex physical systems. This knowledge helps in designing technological devices and understanding natural systems. By understanding these fundamental concepts, you have taken the first step towards further exploration in physics.


Grade 11 → 5.1.3


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Energy in SHM
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Damped and forced oscillations

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