Fowles11

Fowles11 - 1 Chapter 11 11.1 (a) V ( x) = k 2 k2 x + 2 x 2...

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1 Chapter 11 11.1 (a) 2 2 () 2 kk Vx x x =+ 2 2 k Vk x x =− At equilibrium, 0 V ′ = 13 x k = 2 3 2 k x ′′ 23 xk k k = =+ = > 0 Stable (b) ( ) bx k x e = bx bx e b k x e −− At equilibrium 0 bx bx ke bkxe = 1 x b = 2 bx bx bx V bke bkxe b kxe ′′=− + 11 1 1 20 xb Vb k e b k e b k e = + < Unstable (c) 42 2 kx bx 32 Vkx b x At equilibrium kx b x 0 = 0, 2 22 12 2 2 0 2 x b = < 0 Unstable 2 2 62 4 b b k b = 0 > Stable (d) for case (a) 2 3 k m ω = 2 3 3 m Ts k π == = for case (c) at 2 2 2 4 kb m = 2 2 2 4 m kb = 11.2 (, ) 2 4 V x y k x y bx by
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2 2( ) 2) VV kx b ky b xy ∂∂ =− at equilibrium 2 x b and y b == 22 2 20 VVV kk xx y y 2 = k 11 12 21 22 0 2 k k k => = 11 12 2 21 22 40 k > 0 The equilibrium is stabe. 11.3 2 1 () 2 Vx k x dV x dx Fx k x m x dx dx = = = ± ±± ± kx dx mx dx = 0 0 xv x kxdx mx dx = ∫∫ 2 0 kv xx m −= 12 0 dx vk m x x dt 0 0 0 ; xt x dx dt where k m αα ( ) 00 ln ln x x t α +−− = t x xx x e +− = 22 2 2 0 2 tt 2 x e x x e x + cosh 2 ee x t + 11.4 Let the length of the unstretched, elastic cord be d . Then 2 l y 2 dl y = + 2 1 2 2 Vk d lm g y m
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3 ( ) 22 2 44 8 4 2 k Vl y l l y l m =+ −+ + g y ( ) 2 Vk ly l lym g + y The first term, , is an additive constant to the potential energy, so with appropriate adjustment of the zero reference point … 2 4 kl ( ) 2 () 2 2 Vy ky ll y m g y =−+ () 12 22 2 dV y ky y l l y m g dy  =−+ −   At equilibrium, the above expression is zero, so … 4 40 kly ky mg ly −− + = 4 4 kly ky mg −= + 16 16 8 kly km g y mg −+= + 24 3 2 2 2 22 16 8 16 16 8 0 k y kmgy k l m g k l y kl mgy l m g −++ − += 42 2 2 32 44 2 4 2 2 2 0 21 6 21 6 ym g m g m g m g yy y lk l k l k l k l 2 = letting y u l = and 4 mg a kl = 43 2 2 2 ua u a u a −+−+ = 0 11.5 ( ) Vm gh h d φ θ θ b a h 1 h 2 1 cos hb θ = ( ) 2 sin hd ϕ = + ( ) 2 sin cos cos sin ϕθ cos sin ba dd ϕϕ = = 2 sin cos a θϕ = + cos sin gab b θθ =+ + ( ) sin sin cos b b ++
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4 [ ] sin cos Vm g a b θ θθ =− + [ ] cos sin cos g b b a ′′ () 0 g b a Equilibrium … Stable Unstable ab < > 11.6 24 cos 1 ... 2! 4! + 35 sin ... 3! 5! =− + − 1 ... ... 3! 5! gab b   =+ + +− +     3 ... 22 4 a b −− + For a = b, 4 2 ... 12 a + 3 ... 3 mga V ′=− + terms in higher order of 2 ... g a ′′=− + 2 ... g a ′′′=− + 20 g a ′′′′=− < Equilibrium is unstable 11.7 The center of mass (CM) of the hemisphere is 3 8 a from the flat side (see Equation 8.1.8). The height of CM) above the point of contact between the two hemispheres is designated by h 2 in the figure. h 1 is the height of the point of contact above the ground. a φ θ h 1 θ b h 2 ( 12 3 cos cos cos 8 gh h m gb a a ) = + − + ϕ and =
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5 ()( ) 12 3 cos cos 8 b Vm gh h m gab a a θ θθ   =+ = + − +     () 3 sin sin 8 aab ab g ab aa + + =− + + Equilibrium occurs at 0 o = 2 3 cos cos 8 ga b + +    ′′ + +     ( 2 0 3 35 88 m g abba + + + ) 5 a 0 03 Vf o r b >> Therefore, the equilibrium is stable for 3 5 b a < 11.8 From Problem 11.4, we have ( ) 1 2 2 22 2 2 2 2 () 2 2 2 2 1 y V y k y l l y mgy k y l mgy l =−+ =−+− Expanding the square root for small y l 1 2 11 1 1 ...
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This note was uploaded on 03/09/2008 for the course PHYS 301 taught by Professor Mokhtari during the Fall '04 term at UCLA.

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Fowles11 - 1 Chapter 11 11.1 (a) V ( x) = k 2 k2 x + 2 x 2...

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