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phy293_l02.page2

# phy293_l02.page2 - • Capacitor has an electric ﬁeld...

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Changing φ doesn’t change the phase-space ellipse, just the starting point Changing a does change the size of the ellipse, but not the aspect ratio Changing ω 0 changes the aspect ratio (semi-major to semi-minor axes) Careful: The mass is executing a back-and-forth motion (in one dimension: x ), but the phase space in ( x ; ˙ x ) is an ellipse. The equation of the ellipse is: x 2 /a 2 + ˙ x 2 /a 2 ω 2 0 = 1 But the total energy is 1 2 sa 2 = 1 2 2 0 a 2 so can re-write this as: x 2 / (2 E/s ) + ˙ x 2 / (2 E/m ) = 1 or PE ( t ) /E + KE ( t ) /E = 1 or PE ( t ) + KE ( t ) = E The phase space trajectory is an ellipse of constant energy 4. Other Simple Harmonic Oscillation Systems An electrical circuit with an inductor and capacitor is another example Consider the movement of charge around the circuit Inductor has a magnetic field due to the current in the coil: V L = LdI/dt = Ld 2 q/dt 2 = L ¨ q Increasing the current in the coil increases the magnetic field, inducing a voltage in the coil The voltage induced opposes the increasing current (Lenz, Estonia, 1833)
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Unformatted text preview: • Capacitor has an electric ﬁeld created by the charge built up on the top/bottom plates (equal/opposite) • The voltage drop is proportional to the charge: V = q/C • Sum of the voltages must be 0 (Kirchoff, Germany 1845): L ¨ q + q/C = 0 • The solution of ¨ q + ω 2 q = 0 is well known to us, but in this case ω = 1 / √ LC • Total energy: 1 2 LI 2 + 1 2 q 2 /C where the ﬁrst term is magnetic inertia (kinetic) and the second term is electric potential (spring-like) Mass/Spring LC circuit x q F V m L s 1/C 5. Other systems • a) Pendulum: ml ¨ θ + mgθ = 0 and ω = p g/l • b) Torsion Oscillator: I ¨ θ + cθ = 0 and ω = p c/I • e) Water U-tube: ρl ¨ x + 2 gρx = 0 and ω = p 2 g/l • g) Bobber in ﬂuid Torsion: m ¨ x + Agρx = 0 and ω = p AGρ/m • d) Mass on a tensioned string: m ¨ x + 2 Tx/l = 0 and ω = p 2 T/lm...
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