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Physics 104 – Spring 2011. Introduction Syllabus Course MCATs Maglev Tour www.win.net /~ dorsea/nehager/south/atlanta_maglev.htm Differences with Phys. 103 Harmonic Oscillator. American Maglev Technology, Inc. January 2011 Powder Springs, GA. Differences with Phys. 103.
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Physics 104 – Spring 2011 • Introduction • Syllabus • Course • MCATs • Maglev Tour • www.win.net/~dorsea/nehager/south/atlanta_maglev.htm • Differences with Phys. 103 • Harmonic Oscillator American Maglev Technology, Inc. January 2011 Powder Springs, GA
Differences with Phys. 103 • Much more applied • Waves and sound • Thermodynamics • DC circuits • Broader definition of terms (Energy) • Kinetic energy of ideal gas (6.02 x 1023 molecules) • Potential energy in chemical bonds (fuel) • Energy carried by EM wave (ability to do work) • National Energy Policy (managing all this PE and KE)
Harmonic Oscillator • Object subject to restoring force around equilibrium: F = - kx • Force proportional to and opposite displacement • Oscillatory motion around equilibrium • Rate determined by mass m and k • Frictional damping • Examples • Block on a spring (car on springs) • Meter stick anchored one end (diving board) • String in guitar (sound wave) • Object bobbing in water • Molecule in crystal lattice
Energy in Harmonic Oscillator • Potential Energy • Work done by expanding spring: • Work = ∫ -kx dx = (-½ kxf2) - (- ½ kxi2) = ½ kxi2 - ½ kxf2 • Looks like decrease in potential energy • Kinetic Energy • Work gained by block being pushed by spring: • Work = ½ mvf2 - ½ mvi2 • Appears as increase in kinetic energy • Total Energy • Loss of potential = gain in kinetic, and vice-versa • Sum of Kinetic and Potential Constant • E = ½ mv2 + ½ kx2 = ½ kA2 = ½ mvmax2 = const.
Harmonic Oscillator Terminology • Cycle – One complete oscillation • Amplitude - x = -A to x = +A • Period – time to make one cycle • Frequency - # cycles per second • Frequency vs. Period • f = 1/T • T = 1/f
Example E = ½ mv2 + ½ kx2 = ½ kA2 = ½ mvmax2 = const. • 0.3 kg mass (k- 20 N/m) stretched stretched 10 cm and released. • What is maximum kinetic energy? • What is maximum velocity? • What is velocity at 5 cm?
More examples • Problem 15 • What is energy • What is k? • What is m? • What is maximum velocity? • What is velocity at x = .03 m? • Problem 23 • What is energy? • Maximum amplitude? • Maximum acceleration?
Energy of Harmonic Oscillator • At any position E = ½ mv2 + ½ kx2 • At full amplitude: E = ½ mv2 + ½ kx2 = ½ kA2 • At the midpoint E = ½ mv2+ ½ kx2 = ½ mvmax2 • Energy is same for all 3, if you know it for one you know it for others. • Know total energy, subtract for unknown.
Example 11-4 and 11-5 • solve static for spring constant • Amplitude is given • Solve maximum velocity • ½ kA2 = ½ mvmax2 • Magnitude of velocity at x = .05 m • Subtract potential from total energy • Maximum acceleration • Ma = -kx where x is maximum • Energies at x = .05
Vertical Oscillator • Displacement to a new equilibrium balances weight • kΔxoof extra length balances mg at all times. • Oscillation around new equilibrium. • Example 11-1 Car springs kΔxo mg
Equation of Motion • What solves F = ma = -kx ? • If F=ma = constant, then a is constant, see chapter 2 • But a is not constant, a = -kx/m • Try things <?> • SHO Animation • Looks like projection of circular motion. • Specifically sine or cosine. • Substitute sine or cosine • Simple sine cosine – it works • A sinωt or A cosωt - also works • ω = sqrt(k/m) Natural frequency!
Summary to date • Energy E = ½ mv2 + ½ kx2 = ½ kA2 = = ½ mvmax2 • Motion x = A sinωtor x = A cosωt • Harmonic frequency ω = sqrt(k/m) • Frequency and period ω = 2πf ω = 2π/T
