07.Shm And Waves

PRELIMS

1. Simple Harmonic Motion (SHM)

• Periodic Motion: Motion that repeats itself after equal intervals of time.
• Oscillatory Motion: To-and-fro periodic motion about a fixed (mean/equilibrium) position.
• SHM: A special type of oscillatory motion in which the restoring force (or acceleration) is directly proportional to the displacement from the mean position and is always directed towards the mean position.
  • Formula: (or ), where = displacement from mean position, = force constant.
  • Examples: Simple pendulum (for small amplitude), a mass attached to a spring, motion of a tuning fork's prongs.
• Energy in SHM: At the mean position, KE is maximum and PE is minimum (zero); at the extreme positions, PE is maximum and KE is minimum (zero). Total mechanical energy remains constant throughout.

2. Time Period Formulas

• Simple Pendulum: , where = length of the pendulum, = acceleration due to gravity.
  • Seconds Pendulum: A pendulum with a time period of exactly 2 seconds; its length is approximately 99.2 cm (~1 m) at .
  • Time period is independent of the mass of the bob and the amplitude (for small oscillations).
  • At the centre of the Earth, , so the time period becomes infinite (pendulum does not oscillate).
• Spring (Mass-Spring System): , where = mass attached, = spring/force constant.

3. Wave Motion & Basic Properties

• Wave: A disturbance that transfers energy from one point to another without the transfer of matter.
• Wavelength ( ): Distance between two consecutive points in the same phase (e.g., crest to crest, or compression to compression). SI unit: metre.
• Frequency ( or ): Number of oscillations/waves per unit time. SI unit: hertz (Hz).
• Time Period ( ): Time for one complete oscillation; .
• Wave Velocity Relation: (Velocity = Frequency Wavelength).

4. Types of Waves

• Mechanical Waves: Require a material medium to travel (cannot travel through vacuum). E.g., sound waves, waves on strings/water.
• Non-Mechanical (Electromagnetic) Waves: Do not require a medium; can travel through vacuum. E.g., light, radio waves.
• Transverse Waves: Particles of the medium vibrate perpendicular to the direction of wave propagation. E.g., light waves, EM waves, waves on a string; characterized by crests and troughs. Can travel through solids and the surface of liquids, but not through gases/fluids interior (except EM waves, which need no medium at all).
• Longitudinal Waves: Particles of the medium vibrate parallel to the direction of wave propagation. E.g., sound waves; characterized by compressions and rarefactions. Can travel through solids, liquids, and gases.
• Confused Pair: Sound waves are longitudinal and mechanical (need a medium — cannot travel in vacuum); light/EM waves are transverse and non-mechanical (travel fastest in vacuum).

5. Sound Waves

• Speed of Sound: Fastest in solids, slower in liquids, slowest in gases (depends on the elasticity and density of the medium).
  • Speed of sound in air at .
• Audible Range: Human ear can hear frequencies between 20 Hz and 20,000 Hz (20 kHz).
  • Infrasonic: Frequency below 20 Hz (e.g., waves generated by earthquakes).
  • Ultrasonic: Frequency above 20,000 Hz (used in SONAR, medical imaging/ultrasound, cleaning).
• Doppler Effect: Apparent change in frequency of a wave due to relative motion between the source and the observer (e.g., change in pitch of a train's horn as it approaches/recedes).

6. Electromagnetic Waves

• Properties of EM Waves:
  1. They are neutral (uncharged).
  2. They propagate as transverse waves.
  3. They travel at the velocity of light.
  4. They possess both energy and momentum.
• Key Fact: The concept of EM waves was introduced by Maxwell. All EM waves are non-mechanical.

UPSC Relevance

SHM and Wave properties are frequently used to explain natural phenomena and technology.

• Common Trap: Not specifying that EM waves are transverse, while sound waves are longitudinal.
• Mean vs. Extreme: Questions on the energy of a pendulum at different points are a staple of physics conceptual testing.
• Maxwell's Contribution: Often asked in history of science or basic physics identification questions.
• Seconds Pendulum & Centre of Earth: Classic conceptual traps — length of a seconds pendulum (~99.2 cm) and infinite time period at Earth's centre (g=0).
• Sound in Vacuum: A perennial favourite — sound cannot travel in vacuum (mechanical wave) while light/EM waves can and do (this explains why explosions in space are silent).
• Doppler Effect: Real-world tie-in with train horns, and its application in radar/SONAR and astronomy (redshift).