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Periodic Motion and Theory of Oscillations

Periodic Motion and Theory of Oscillations. a x. Harmonic oscillator: ma x = - kx. Restoring force F x = -kx is a linear function of displacement x from equilibrium position x=0. m. 0. Oscillator equation:. X. Initial conditions at t=0:. Simple harmonic motion:

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Periodic Motion and Theory of Oscillations

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  1. Periodic Motion and Theory of Oscillations ax Harmonic oscillator: max = - kx Restoring force Fx = -kx is a linear function of displacement x from equilibrium position x=0. m 0 Oscillator equation: X Initial conditions at t=0: Simple harmonic motion: Position, velocity, and acceleration are periodic, sinusoidal functions of time.

  2. Energy in Simple Harmonic Motion Total mechanical energy E=K+U in harmonic oscillations is conserved: Example:Non-adiabatic perturbation of mass (a) M → M + m at x=0 results in a change of velocity due to momentum conservation: Mvi=(M+m)vf, vf= Mvi/(M+m), hence, Ef= MEi/(M+m), Af= Ai[M/(M+m)]1/2, Tf = Ti [(M+m)/M]1/2 (b) M → M + m at x=A (v=0) does not change velocity, energy, and amplitude; only the period is changed again due to an increase of the total mass Tf = Ti [(M+m)/M]1/2

  3. Exam Example 30: A Ball Oscillating on a Vertical Spring(problems 14.38, 14.83) y Data: m, v0 , k y2=y0+A • Find:(a) equilibrium position y0; • (b) velocity vy when the ball is at y0; • amplitude of oscillations A; • (d) angular frequency ω and • period T of oscillations. Unstrained→ 0 v0 y0 Equilibrium Solution: Fy = - ky • Equation of equilibrium: • Fy – mg = 0, -ky0 = mg , y0 = - mg/k • (b) Conservation of total mechanical energy Lowest position y1=y0-A v1=0 (c) At the extreme positions y1,2 = y0 ± A velocity is zero and (d)

  4. Applications of the Theory of Harmonic Oscillations Oscillations of Balance Wheel in a Mechanical Watch (mass m) Newton’s 2nd law for rotation yields R Exam Example 31: SHM of a thin-rim balance wheel(problems 14.41,14.97) Data: mass m, radius R , period T Questions: a) Derive oscillator equation for a small angular displacement θ from equilibrium position starting from Newton’s 2nd law for rotation. (See above.) b) Find the moment of inertia of the balance wheel about its shaft. ( I=mR2 ) c) Find the torsion constant of the coil spring.

  5. Vibrations of Molecules due to van der Waals Interaction Potential well for molecular oscillations m m Displacement from equilibrium x = r – R0 Restoring force Approximation of small-amplitude oscillations: |x| << R0 , (1+ x/R0)-n ≈ 1 – nx/R0, Fr = - kx , k = 72U0/R02 Example: molecule Ar2 , m = 6.63·10-26kg, U0=1.68·10-21 J, R0= 3.82·10-10 m

  6. Simple and Physical Pendulums Newton’s 2nd law for rotation of physical pendulum: Iαz = τz , τz = - mg d sinθ ≈ - mgd θ Simple pendulum: I = md2 Example: Find length d for the period to be T=1s.

  7. Exam Example 32: Physical Pendulum (problem 14.99, 14.54) 0 X Data:Two identical, thin rods, each of mass m and length L, are joined at right angle to form an L-shaped object. This object is balanced on top of a sharp edge and oscillates. m m θ d Find:(a) moment of inertia for each of rods; (b) equilibrium position of the object’s center of mass; (c) derive harmonic oscillator equation for small deflection angle starting from Newton’s 2nd law for rotation; (d) angular frequency and period of oscillations. cm y Solution: (a) dm = m dx/L , (b) geometry and definition xcm=(m1x1+m2x2)/(m1+m2)→ ycm= d= 2-3/2 L, xcm=0 (c) Iαz = τz , τz = - 2mg d sinθ ≈ - 2mgd θ (d) Object’s moment of inertia

  8. Damped Oscillations Springs in the automobile’s suspension system: oscillation with ω0 The shock absorber: damping γ

  9. Damped Oscillations Frictional force f = - b vx dissipates mechanical energy. Newton’s 2nd law: max = -kx - bvx Differential equation of the damped harmonic oscillator: Fourier analysis: General solution: underdamped (γ < γcr) (instability if γ<0) overdamped (γ > γcr) Critical damping γcr = ω0 , bcr=2(km)1/2 Damping power:

  10. Forced Oscillations and Resonance Forced oscillator equation: Amplitude of a steady-state oscillations under a sinusoidal driving force F = Fmax cos(ωdt) At resonance, ωd ≈ ω, driving force does positive work all the time Wnc = Ef – Ei >0, and even weak force greatly increases amplitude of oscillations. Example: laser ( ←→ )→ (self-excited oscillation of atoms and field) Parametric resonance is another type of resonance phenomenon, e.g. L(t).

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