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Understanding Simple Pendulum and Electromagnetic Oscillations

This document explores the principles of simple pendulum motion and electromagnetic oscillations, detailing the restoring forces and quantitative analyses of energy in L-C circuits. It emphasizes the relationship between charge, current, and voltage in oscillating circuits, explaining phase differences in oscillatory motion. Additionally, it discusses the superposition of simple harmonic motions (SHMs), showcasing how multiple oscillations combine, and the effects of phase differences on resultant amplitudes. A thorough breakdown is provided to facilitate understanding of oscillatory systems.

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Understanding Simple Pendulum and Electromagnetic Oscillations

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  1. O  l T mg 12-5 Simple pendulum (p307): Restoring force: Comparing to SHM

  2. 28-5 Electromagnetic Oscillation (P650) 1. Oscillation circuit Inertial element of current: L L─ C circuit Restoring element of voltage: C Qualitative explanation Electric energy in C Magnetic energy in L Quantitative analysis: emf of self-induction on L= Voltage cross C Since Comparing to q is a SHQ.

  3. 2. Discussion: ① Since q is a SHQ, I is a SHQ. I is ahead of q for phase of /2. ③ U is a SHQ. U and q are in phase. Electric energy : ④ Magnetic energy: Total energy: (Conserved）

  4. # Superposition of Oscillations x = x1+ x2 x=Acos( t+) That is, 1. Superposition of Two SHMs in Same Direction with Same Frequency(二同向、同频简谐运动合成): x1=A1cos( t+ 1) x2=A2cos( t+ 2) The resultant oscillation is also a SHM in Same Direction with Same Frequency.

  5. ① If f f2 f1 The resultant oscillation is related to the phase difference f2-f1 . Discussion: 1). Two special cases: =A1+A2 —–When the two SHMs are in phase, the resultantamplitude reaches its maximum.

  6. 2).In general cases:, ② If A=|A1-A2| —–When the two SHMs are out of phase, the resultantamplitude reaches itsminimum. If A1=A2, then A=0 ! 3). The phase difference of two SHMs plays an important role in the resultant oscillation ! 4). In similar way, we may get the results for the superposition of multi-oscillations .

  7. 2. Superposition of N SHMs in Same Direction with Same Amplitude,Same Frequency and SameAdjacent Phase Difference(N同向、同幅、同频、同相邻位相差谐振动合成): N C C b N C R Nb 1 R a A O 1 0 O

  8. Example: x=? Method1: 2 x12 and x3:

  9. Method 2. 3 SHMs in Same Direction with Same Amplitude,Same Frequency and SameAdjacentPhase Difference Method 3: Rotating Vector:

  10. Questions (思考题） • P316 7; P316 11; P316 14 • Problems (练习题） • P320 46; P322 76; P323 85

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