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 ↓


Equations of SHM


Simple harmonic motion (SHM) is a type of periodic motion in which an object oscillates back and forth from an equilibrium position. This motion is characterized by its sinusoidal nature, which means it can be described using sine and cosine functions. It is essential in understanding various physical systems, from pendulums to springs.

What is simple harmonic motion?

Imagine there is a spring attached to the wall. When you pull and release it, it oscillates back and forth. This is a classic example of SHM. The motion can be described as:

  • Periodic: It is repeated at regular intervals.
  • Sinusoidal: It follows the sine or cosine curve.

In simple terms, SHM is a repeated motion around an equilibrium or focal point. The object moves back and forth within a fixed distance on each side of the equilibrium position.

Equations of SHM

To understand SHM mathematically, we represent the motion using equations. The elementary equations of SHM are:

Displacement equation

The displacement of an object in SHM at any time t can be represented as:

x(t) = A cos(ωt + φ)

Where:

  • A = amplitude (maximum displacement from equilibrium)
  • ω = angular frequency (in radians per second)
  • t = time (in seconds)
  • φ = phase angle (in radians)

Velocity equation

The velocity of the object can be obtained by differentiating the displacement with respect to time:

v(t) = -Aω sin(ωt + φ)

Acceleration equation

Acceleration is obtained by differentiating the velocity equation with respect to time:

a(t) = -Aω 2 cos(ωt + φ)

Note that both velocity and acceleration are periodic functions with time.

Visualization of SHM

x(t) balance An object oscillates along the x-axis in SHM.

In the SVG example above, the red circle represents an object moving in SHM along the x-axis. The line shows its path, with the midpoint representing the equilibrium position.

Breaking down the equations

Dimensions (A)

Amplitude is the peak value of displacement, which represents the maximum distance moved by the object from the equilibrium position. It represents the spread or extent of oscillation.

Example: If a pendulum swings 30 cm in either direction from its central position, then its amplitude is 30 cm.

Angular frequency (ω)

Angular frequency describes how quickly the oscillations occur. Technically, it is the rate at which the object completes one cycle of oscillation, expressed in radians per second.

Example: If a mass on a spring completes ten rotations in one second, then its angular frequency will be high.

Phase angle (φ)

The phase angle helps to determine the initial conditions of SHM. It indicates where the oscillation starts at t=0.

Example: Phase angle 0 means the motion starts from the farthest point of positive displacement.

Exploring equations with examples

Let's use some examples to make these equations more clear:

Example 1: Spring oscillator

Consider a mass attached to a spring on a smooth surface. If the spring is compressed and released, the mass oscillates with SHM. Suppose the amplitude is 0.5 m, the angular frequency is 2 rad/s, and the phase angle is 0 rad.

The displacement can be given as follows:

x(t) = 0.5 cos(2t)

The velocity of the mass will be:

v(t) = -0.5 * 2 sin(2t) = -1 sin(2t)

and the acceleration is:

a(t) = -0.5 * (2 2) cos(2t) = -2 cos(2t)

Consider this hypothetical spring-mass system operating in orbit:

You stretch an initially 20 cm spring by 5 cm. The system exhibits SHM with:

  • Dimensions, A = 0.05 m
  • Angular frequency, ω = 3 radian/second
  • Phase angle, φ = 0

The equations are as follows:

Displacement: x(t) = 0.05 cos(3t)

Velocity: v(t) = -0.15 sin(3t)

Acceleration: a(t) = -0.45 cos(3t)

Example 2: Simple pendulum

A pendulum swinging with a small angle theta can also be modeled as SHM. Suppose its amplitude is 0.1 radians, with no initial phase shift and a period of 2 seconds. First, calculate its angular frequency using the formula:

ω = (2π) / T

where T is the period. Insert our values:

ω = (2π) / 2 = π rad/s

Now substitute this into the displacement equation with A = 0.1 radians and φ = 0:

x(t) = 0.1 cos(πt)

Thus, the displacement of the pendulum at any time t can be described by this equation. Similarly, you can find the velocity and acceleration.

This type of modeling provides practical information about the motion dynamics of such simple devices.

Graphical representation

The key to visualizing SHM is to understand how these mathematical quantities correspond to physical motion. This section presents a step-by-step guide to plotting the SHM function.

Using the displacement, velocity, and acceleration formulas of SHM, we can plot their corresponding waveforms over time, which is educational and practical. Here's how it works:

  • The displacement graph is a cosine wave that starts with a maximum value (amplitude).
  • The velocity graph is a sine wave with respect to time and is shifted by π/2.
  • The acceleration graph is a cosine wave, but inverted and larger due to the factor -Aω 2 .
Tea x(t) Displacement Graphical plot of displacement in SHM.

Important considerations

As you delve deeper into SHM, it will be necessary to consider the underlying assumptions:

  • The restoring force must always be proportional to the negative of the displacement (Hooke's law).
  • SHM is an idealization; in real systems, factors such as friction and air resistance alter the motion.
  • SHM formulas are mainly applicable for very small angles/oscillations, where accuracy is not at stake.

These aspects shed light on why SHM remains fundamental to mastering classical physics and its practical applications.

Real life applications of SHM

Understanding SHM broadens the horizon of connecting and applying physics concepts to real-world systems:

Wrist watch

Mechanical wristwatches use balance wheels, where the harmony of oscillations ensures accuracy in timing.

Seismology

Seismographs help measure earthquakes by using simple harmonic oscillators as the main component to sense and display the Earth's vibrations.

Musical instruments

The strings in piano and guitar vibrate with SHM, producing the desired tones and pitches.

Recognizing the principles of SHM in everyday life provides remarkable insights beyond classroom theory, inviting us to appreciate the dynamic that lies at the core of natural phenomena.

Conclusion

We have covered the fundamentals of simple harmonic motion and its mathematical equations. By understanding the displacement, velocity, and acceleration equations, you will gain a deeper understanding of the behavior of oscillating systems in nature.

Moving forward, exploring practical applications of SHM enables you to effectively connect the principles of physics with essential day-to-day systems, from technologies to natural phenomena. Finally, remember, understanding concepts like SHM helps simplify complex motions, which positively positions your understanding of physics in a real-world setting.


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Equations of SHM

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