Physics XI - Chapter 14: Oscillations

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Quick Revision Box: Oscillations

  • Oscillation: A repeated back-and-forth motion about a mean position. E.g., a swinging pendulum.
  • Period (T): Time taken for one complete oscillation. SI unit: second (s).
  • Frequency (ν): Number of oscillations per second. SI unit: Hertz (Hz). Relation: ν = 1/T.
  • Displacement (x): Instantaneous distance from the mean position.
  • Amplitude (A): Maximum magnitude of displacement from the mean position.
  • Simple Harmonic Motion (SHM): Oscillation where restoring force is directly proportional to displacement and acts towards the mean position. F = -kx.
  • SHM Equation: x = A sin(ωt + φ) or x = A cos(ωt + φ).
  • Angular Frequency (ω): ω = 2πν = 2π/T. Relates to how fast oscillation occurs.
  • Phase (ωt + φ): Argument of the sine/cosine function that determines the state of oscillation.
  • Phase Constant (φ): Initial phase at t=0. Depends on initial conditions.
  • Velocity in SHM: v = ω√(A² - x²). Velocity is maximum at mean position (v_max = ωA) and zero at extremes.
  • Acceleration in SHM: a = -ω²x. Acceleration is always directed towards the mean position and is maximum at extremes (a_max = ω²A).
  • Total Energy: Total mechanical energy is constant. E = (1/2) kA² = (1/2) mω²A².
  • Spring-Mass System: ω = √(k/m) and T = 2π√(m/k).
  • Simple Pendulum: For small angles, T = 2π√(L/g). Period is independent of the mass of the bob.
  • Free Oscillation: Natural oscillation with a fixed frequency (ω) determined by the system's properties (m, k, L).
  • Damped Oscillation: Oscillation where amplitude decreases with time due to resistive forces.
  • Forced Oscillation & Resonance: When a system is driven by a periodic force. Resonance occurs when the driving frequency matches the system's natural frequency, leading to a maximum amplitude.

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Chapter Summary

This chapter introduced us to the beautiful and predictable world of Oscillations, the to-and-fro motions we see in a swing, a guitar string, or a building in an earthquake. We began by defining key terms like period, frequency, and amplitude, which are the fundamental language for describing any periodic motion.

The heart of the chapter is Simple Harmonic Motion (SHM). We learned that SHM is not just any oscillation; it's a specific type where the restoring force is directly proportional to the negative of the displacement (F = -kx). This leads to the characteristic sine and cosine equations. We dived deep into the kinematics of SHM, deriving and understanding the expressions for velocity and acceleration, and how they vary with displacement. A key takeaway is that energy in SHM is conserved, continuously transforming between kinetic and potential forms.

Finally, we applied these concepts to real systems like the spring-mass system and the simple pendulum, learning how to calculate their natural periods. We concluded by exploring what happens in the real world with damped oscillations and the dramatic phenomenon of resonance, where a small periodic force can produce large amplitude vibrations, a crucial concept in engineering and design.

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