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 ↓


Energy in SHM


Simple harmonic motion (SHM) is a type of oscillatory motion. When a system undergoes SHM, it moves repeatedly back and forth from its equilibrium position. Two key concepts to understand in SHM are potential energy and kinetic energy. These energies help explain how and why an object oscillates in SHM.

Basics of Simple Harmonic Motion

SHM occurs when the force applied to an object moves it toward its equilibrium position and is proportional to the displacement from that position. This often happens with systems like springs or pendulums.

Equation of motion

The motion of a simple harmonic oscillator can be described by the following equation:

F = -kx

Here, F is the restoring force, k is the spring constant, and x is the displacement from the equilibrium position.

Energy in SHM

In SHM, energy is transferred between kinetic and potential forms. At maximum displacement, all the energy is potential. At equilibrium, all the energy is kinetic.

Kinetic Energy (KE)

Kinetic energy is the energy an object has because of its motion. In SHM, when an object is moving at its fastest speed, its kinetic energy is maximum. The formula for kinetic energy is:

KE = (1/2)mv²

Here, m is the mass, and v is the velocity.

Potential Energy (PE)

In SHM, potential energy is the stored energy that arises due to the change in the state of the system. For a spring, the potential energy is given by:

PE = (1/2)kx²

Here, k is the spring constant, and x is the displacement from equilibrium.

The total mechanical energy

The sum of the kinetic and potential energy of a system undergoing SHM remains constant if there are no other forces acting on it like friction. It is expressed as:

E = KE + PE = constant

This implies that the total energy is conserved with time.

Illustration of energy transformation

Let us visualize how the energy transition between kinetic and potential energy occurs in SHM by taking the example of a simple pendulum. In this representation, the levels of kinetic energy and potential energy change but their total remains the same.

Picture a pendulum swinging back and forth. At its highest point, the pendulum's entire energy is potential. At its lowest point, it is entirely kinetic.

High P.E. High K.E.

Examples of energy conservation

The mass-spring system is another classic example of SHM. Imagine a mass attached to a spring. When you pull and release the mass, you can see the transformation of energy:

  • When you pull on a mass, it stores energy as potential energy (PE = (1/2)kx²).
  • As the mass passes through the balance, the energy is converted into kinetic energy (KE = (1/2)mv²).
  • Beyond equilibrium, the kinetic energy is converted back into potential energy as it compresses or extends the spring in the opposite direction.

During this oscillation the total energy of the system remains constant.

Mathematical insights on energy in SHM

Let us find out how the energy changes with time mathematically. In a perfect harmonic oscillator,

The equation of displacement is:

x(t) = A cos(ωt + φ)

Here, A is the amplitude, ω is the angular frequency, and φ is the phase angle. The velocity v(t) is given by differentiating x(t) with respect to time t:

v(t) = -Aω sin(ωt + φ)

Substituting this into the kinetic energy equation gives:

KE(t) = (1/2)m(Aω sin(ωt + φ))²

For potential energy:

PE(t) = (1/2)k(A cos(ωt + φ))²

Remember, in a simple harmonic oscillator k = mω².

Practical examples and applications

Real-world examples where SHM and its energy calculations are used include:

  • Seismic Waves: Using the principles of SHM to understand the motion observed in earthquakes.
  • Pendulum clocks: The exchange of potential and kinetic energy allows the pendulum to keep accurate time by maintaining a constant oscillation period.
  • Car suspension system: Energy conversion in springs and shock absorbers makes the vehicle's travel comfortable.

Real-world challenges

The complexities in real-world SHM applications are:

  • Damping: Friction or air resistance can dissipate energy, causing the oscillations to slowly decay without external energy input.
  • Non-harmonic forces: Real systems may not obey Hooke's law exactly and may have additional forces that alter the motion.

Conclusion

Energy in SHM is a fascinating study of how potential and kinetic energies interconvert over time while conserving total mechanical energy in an ideal scenario. Understanding the energy dynamics within SHM not only clarifies simple oscillatory systems but also enhances the understanding of more complex, real-world phenomena.


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Equations of SHM
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Energy in SHM

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