(7)+Oscillations

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= = =Oscillation - motion that repeats itself =
 * ** Dampened Oscillation ** - motion which dies out


 * ** Forced Oscillation - ** energy is replenished by outside force in order to maintain motion


 * ** Simple Harmonic Motion (SHM) ** - restoring force is proportional to displacement. Motion is cyclic and can be described using a sinusoidal curve


 * Amplitude (A) - max displacement from equilibrium point (same as x for a spring) (m)


 * Period (T) - time it takes to complete one cycle (s)

Due to Taylor Expansion: for small angles ( θ < 10 degrees) sinθ approximately equals θ Therefore, F (restoring) approx = mgθ
 * Frequency (f) - number of cycles per second (Hz or 1/s)

V = dx/dt = -Aω sin(ωt)
Vmax = (+/-) Aω

a = dv/dt = -Aω^2 cos(ωt) a max = (+/-) Aω^2 a(t) = -ω^2 [x(t)]

ΣF max = (m)(a max)
-kx = (m)(a max) a max = -(k/m)(A) -kx = m(-ω^2)(x) ω = (k/m)^1/2 ω = 2π/T = (k/m)^1/2 Tsp = 2π (m/k)^1/2

KE = 1/2mv^2 V = -Aω sin(ωt) K(t) = 1/2m[(ω^2)(A^2)(sin^2(ωt))] Usp = 1/2kx^2 U(t) = 1/2k[(A^2)(cos^2(ωt))]
 * Kinetic Energy

**

Σ τ = Iα mgsin θ(L) = Iα α = [mgsin θ(L)] / I (for small angles, sinθ = θ) so α = [mg θ(L)] / I ω = 2π/T = [mg L / I] ^1/2 Tpp = 2π [ I / mgL] ^1/2 (where L = h com = distance from the system's COM to the axis of rotation) I = mL^2 Tp = 2π [ mL^2 / mg L] ^1/2 Tp = 2π [ L/g ] ^1/2
 * Period of a physical pendulum: **
 * Period of a simple pendulum: **



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http://en.wikipedia.org/wiki/File:Simple_gravity_pendulum.svg http://media.ehs.uen.org/html/PhysicsQ3/Total_Energy_of_the_Oscillation_01/pend1.png http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Dynamics/Forces/SpringForce.html