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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.
 Fall '09
 Hicks
 Thermodynamics

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