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


Introduction to simple harmonic motion

Motion is all around us. From the swinging of a pendulum to the vibration of guitar strings, motion can be observed in various forms. One of these forms is known as simple harmonic motion (SHM), which is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.

Imagine a child sitting on a swing. As the child swings back and forth, he is experiencing an example of simple harmonic motion. The swing swings back and forth in a regular pattern, showing the characteristics of SHM.

Defining simple harmonic motion

Simple harmonic motion can be defined as:

Simple harmonic motion (SHM) is a type of oscillatory motion in which the restoring force acting on an object is proportional to the displacement of the object from its equilibrium position, and is directed towards that position.

Mathematically this can be expressed as:

F = -kx

Here, F is the restoring force, k is the proportionality constant (spring constant in most cases), and x is the displacement from the equilibrium position. The minus sign indicates that the direction of the force is opposite to the direction of displacement.

Characteristics of simple harmonic motion

Simple harmonic motion has specific characteristics that can be observed in different physical systems. These characteristics are:

1. Periodicity

SHM is a periodic motion, which means that the object returns to its initial position after a certain time interval, known as the period.

2. Sinusoidal nature

SHM can be described by sinusoidal functions (sine and cosine functions). This means that the displacement of the object as a function of time can be expressed using these functions.

The general form of the equation for SHM is:

x(t) = a cos(ωt + φ)

Where:

  • x(t) is the displacement at time t.
  • A is the amplitude of the motion, the maximum displacement from the equilibrium position.
  • ω (omega) is the angular frequency.
  • φ (phi) is the phase constant, which determines the initial condition of the motion.

3. Oscillation

The motion of an object consists of back and forth motion about its equilibrium position.

Mathematical description of simple harmonic motion

Angular frequency and period

The angular frequency ω is related to the physical frequency f and the period T as follows:

ω = 2πf = 2π/t

Here:

  • T is the period (time taken in one complete cycle of the motion).
  • f is the frequency (number of cycles per second).

Velocity and acceleration in SHM

In SHM the velocity of an object is the time derivative of its displacement:

v(t) = dx/dt = -Aω sin(ωt + φ)

The acceleration of the object is the time derivative of velocity, or the second derivative of displacement:

a(t) = dv/dt = d²x/dt² = -Aω² cos(ωt + φ)

Energy in simple harmonic motion

In simple harmonic motion energy is continuously transformed between potential energy and kinetic energy.

Kinetic energy

The kinetic energy KE of a body in SHM can be described as:

KE = 1/2 m v² = 1/2 m (aω sin(ωt + φ))²

Potential energy

The potential energy PE is:

PE = 1/2 k x² = 1/2 k (A cos(ωt + φ))²

Total energy

In SHM, the total mechanical energy E remains constant:

E = KE + PE = 1/2 k A²

This energy transformation results in the characteristic sinusoidal motion of SHM.

Visualization of simple harmonic motion

To understand SHM better, let's visualize it using the example of a simple pendulum:

equilibrium position

In the figure above, the pendulum swings from side to side, which shows SHM. The equilibrium position is just below the pivot point.

Examples of simple harmonic motion

Example 1: Mass on a spring

Consider a mass attached to a spring. When the mass is displaced from its equilibrium position and released, it will oscillate back and forth. This is a classic example of SHM.

Example 2: Simple pendulum

A simple pendulum exhibits SHM when the angle of swing is small. The acceleration due to gravity acts as a restoring force that pulls the pendulum back toward its equilibrium position.

Example 3: Vibrations of a tuning fork

When a tuning fork is struck, its prongs vibrate in SHM, producing sound waves with a specific pitch.

Importance of simple harmonic motion

Understanding SHM is important in the study of waves and oscillations because it forms the basis of more complex motions and systems. It provides information about the behavior of various physical phenomena, including sound waves, light waves, and electrical circuits.

Conclusion

Simple harmonic motion is a fundamental concept in physics, representing an idealized motion that underlies a wide range of real-world systems. By studying SHM, we gain a deeper understanding of how forces interact to produce recurring patterns of motion.


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Simple Harmonic Motion

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