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 ↓


Pendulum and spring-mass system


In physics, the study of waves and oscillations is fundamental to understanding the underlying principles of many phenomena. An important part of this study is simple harmonic motion, or SHM for short. SHM describes the kind of oscillatory motion where the force acting on an object is proportional to its displacement from its equilibrium position and acts toward that equilibrium. The two most compelling systems that demonstrate SHM are the pendulum and the spring-mass system.

Simple pendulum

A simple pendulum consists of a mass, known as the bob, attached to a string of length (L) that swings back and forth under the influence of gravity. This is a classic example of SHM when the angle of swinging is small.

Components of a pendulum

  • Bob: A mass at the end of a pendulum.
  • String: length ( L ), considered weightless and rigid.
  • Pivot point: The fixed point from which the pendulum swings.

Mathematics of SHM of a pendulum

When the pendulum is displaced from its rest position, there is a restoring force on it due to gravity. This force brings it back to the equilibrium position. For small angles (less than about 15 degrees), this force can be considered proportional to the displacement, leading to a simple harmonic oscillator.

The equation of motion of a pendulum is given as:

[theta''(t) + frac{g}{L} sin(theta(t)) = 0]

For small angles, ( sin(theta) approx theta ), and the equation simplifies to:

[theta''(t) + frac{g}{L} theta(t) = 0]

The solution to this differential equation is:

[theta(t) = theta_0 cosleft(sqrt{frac{g}{L}} t + phiright)]

Where:

  • (theta_0) is the maximum angle of displacement.
  • (g) is the acceleration due to gravity.
  • (phi) is the phase constant, determined by the initial conditions.

Characteristics of simple pendulum motion

  • Period (T): The period of a simple pendulum is the time it takes to complete one full cycle of its motion. It is given as: 
    [T = 2pi sqrt{frac{L}{g}}]
    Note that the time period is independent of the mass of the bob and the amplitude of the motion (for small angles).
  • Frequency ( f ): Frequency is the reciprocal of the period: 
    [f = frac{1}{T} = frac{1}{2pi} sqrt{frac{g}{L}}]
  • Amplitude: Displacement from central position.
l (theta)

Spring-mass system

Another classical system that exhibits simple harmonic motion is the spring-mass system. This system consists of a mass attached to a spring that can oscillate back and forth.

Components of a spring-mass system

  • Mass: The object at the end of a spring.
  • Spring: An elastic object that can be compressed or stretched.

Mathematics of spring-mass SHM

The force exerted by a spring is given by Hooke's law, which states that the force exerted by a spring is proportional to the distance it is stretched or compressed from its rest position:

[F = -kx]

Where:

  • (F) is the force exerted by the spring.
  • (k) is the spring constant, which measures the stiffness of the spring.
  • (x) is the displacement of the spring from its equilibrium position.

The equation of motion for a mass attached to a spring is given by Newton's second law:

[ma = -kx]

where (a) (acceleration) is the second derivative of displacement with respect to time, so

[mfrac{d^2x}{dt^2} = -kx]

On rearranging we get:

[frac{d^2x}{dt^2} + frac{k}{m}x = 0]

The general solution is this:

[x(t) = A cosleft(omega t + phiright)]

Where:

  • (A) is the amplitude, is the maximum extent of displacement.
  • (omega = sqrt{frac{k}{m}}) is the angular frequency.
  • (phi) is the phase constant, determined by the initial conditions.

Features of the motion of the spring-mass system

  • Period ( T ): The period is given by: 
    [T = 2pi sqrt{frac{m}{k}}]
  • Frequency ( f ): The frequency is: 
    [f = frac{1}{T} = frac{1}{2pi} sqrt{frac{k}{m}}]
  • Amplitude: Maximum displacement from equilibrium.
M K

Comparison of pendulum and spring-mass system

Both pendulum and spring-mass systems are examples of SHM, yet they exemplify these principles in different ways.

Similarities

  • Both systems have a state of equilibrium to which they naturally return when disturbed.
  • Both exhibit periodic motion and have a definite period and frequency.
  • In both systems there is energy conversion between potential and kinetic energy during motion.

Contraindications

  • The restoring force in a pendulum is due to gravity, while in a spring-mass system it is due to the spring tension.
  • The time period of a pendulum depends on the length of the string and gravity, while the time period of a spring-mass system depends on the mass and the spring constant.

Conclusion

The study of simple harmonic motion through pendulum and spring-mass systems provides insight into fundamental aspects of oscillatory motion. Understanding these systems lays the groundwork for further exploration into more complex forms of motion and wave phenomena. Despite the simplicity of these models, they are powerful tools in understanding the physical world and are a starting point for exploring the vast realm of oscillations and waves in physics.


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Pendulum and spring-mass system

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