# Part c change g to g9g wmg9w just use t lg

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Unformatted text preview: y, September 30, 2013 Pendulum by DA, cont. • Original pendulum has M=?, L=?, T=1 sec. • Part (a): replace M with M’ (also =?). • Part (b): replace L with L’=4 L. • Part (c): change g to g’=9g. (W’=mg’=9W.) ￿ • Just use T ∼ (L/g) Immediately gives: • (a) T’=1 sec, (b) T’=2 sec, (c) T’=1/3 sec. Monday, September 30, 2013 DA: use M, L, T for all [F orce] = [ma] = M L/T 2 1N = 1kg m/s2 2 2 [Energy ] = [mv ] = M L /T 2 [P ower] = [E/T ] = M L2 /T 3 2 [P ressure] = [F/A] = M/T L 2 2 [T emp.] = [E/kB ] = M L /T kB Monday, September 30, 2013 2 1J = 1kg m /s 2 1W = 1kg m2 /s3 2 2 1P a = 1N/m = 1kg/s m We’ll discuss Temp more soon. Appetizer: Review SHO • Simple setup, applies all over the place in Nature. Whenever small displacement from equilibrium. Ignore friction here. Consider motion in 1d, along x axis. Let x=0 be the equilibrium position. Force F(x) with F(0)=0. For small x, can always Taylor expand F(x) and keep just the ﬁrst term: F (x) ≈ −kx 2 [k ] = [F/L] = [M/T ] Monday, September 30, 2013 Physics of the SHO Monday, September 30, 2013 Now the math: key: - sign for oscillations 2 dx m 2 = −kx dt x(t) = A cos(ω t + φ) 2 dx 2 = −ω x dt2 ￿ ω ≡ k/m −1 [ω ] = [1/T ] = s ( A= “amplitude”, phi = “phase”, determined e.g. by initial position and velocity of the mass. ). x( t + T ) = x( T ) T = 2π /ω = 2π *(DA!) Monday, September 30, 2013 ￿ m/k * SHO energy 1 2 Recall: E = mx + V (x) Find potential: ˙ 2 1 dV 22 2 V ( x ) = mω x F =− = − mω x 2 dx Plug in x(t) = A cos(ω t + φ) get E=constant: 1 22 E = mω A 2 KE: Double check units: 2 2 [E ] = [mv ] = [mω A ] 1 1 2 mx = mω 2 A2 sin2 (ω t + φ), ˙ 2 2 Monday, September 30, 2013 2 PE: 1 1 22 mω x = mω 2 A2 cos2 (ω t + φ), 2 2 Any small oscillation system: same math * E.g. pendulum: d2 θ * 2 mL = −mgL sin θ ≈ −mgLθ dt2 2 dθ 2 ≈ −ω θ dt2 θ(t) = θ0 cos(ω t + φ) ￿ ω = g/L ￿ Advice: review this for T = 2π /ω = 2π L/g (DA!) Monday,...
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## This note was uploaded on 02/11/2014 for the course PHYSICS 2C taught by Professor Hicks during the Fall '09 term at UCSD.

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