By Fowles G.R., Cassiday G.L.
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Extra info for Analytical mechanics, 7ed., Solutions manual
22 Choose a coordinate system with the origin at the center of the wheel, the x′ and y′ axes pointing toward fixed points on the rim of the wheel, and the z ′ axis pointing toward the center of curvature of the track. Take the initial position of G the x′ axis to be horizontal in the −VD direction, so the initial position of the y′ axis is vertical. V The bicycle wheel is rotating with angular velocity D about its axis, so … b V G ωl = kˆ′ D b A unit vector in the vertical direction is: Vt Vt nˆ = iˆ′ sin D + ˆj ′ cos D b b At the instant a point on the rim of the wheel reaches its highest point: Vt Vt G r ′ = bnˆ = b iˆ′ sin D + ˆj ′ cos D b b Since the coordinate system is moving with the wheel, every point on the rim is fixed in that coordinate system.
15c to find the time it takes the projectile to strike the ground … 58 1 z ′ ( t ) = − gt 2 + vD′ t sin α + ω vDt 2 cos α cos λ = 0 2 2vD′ sin α 2v′ sin α or t= ≈ D g − 2ω vD′ cos α cos λ g We have ignored the second term in the denominator—since vD′ would have to be impossibly large for the value of that term to approach the magnitude g g − 2ω vD′ cos α cos λ ≈ g − ω vD′ For example, for λ = 41o and α = 45o or vD′ ≈ g ω ≈ 144 km ! 22 Choose a coordinate system with the origin at the center of the wheel, the x′ and y′ axes pointing toward fixed points on the rim of the wheel, and the z ′ axis pointing toward the center of curvature of the track.
22 In steady state, x ( t ) = ∑ An e ( i nω t −φn ) n An = Fn m 1 (ω 2 − n 2ω 2 )2 + 4γ 2 n 2ω 2 2 4F , n = 1,3,5, . . 24 The equation of motion is F ( x ) = x − x 3 = mx . For simplicity, let m=1. Then (a) (b) x = x − x 3 . This is equivalent to the two first order equations … x = y and y = x − x3 The equilibrium points are defined by x − x 3 = x (1 − x )(1 + x ) = 0 Thus, the points are: (-1,0), (0,0) and (+1,0). We can tell whether or not the points represent stable or unstable points of equilibrium by examining the phase space plots in the neighborhood of the equilibrium points.
Analytical mechanics, 7ed., Solutions manual by Fowles G.R., Cassiday G.L.