Part c change g to g9g wmg9w just use t lg

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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,...
View Full Document

This note was uploaded on 02/11/2014 for the course PHYSICS 2C taught by Professor Hicks during the Fall '09 term at UCSD.

Ask a homework question - tutors are online