2 Prescribed Motion And Dynamic Equation 190 Points A 6 Kg Collar Travels Along Portoctly Smooth Horizontal Rod Defi 1 (35.78 KiB) Viewed 20 times
2 Prescribed Motion And Dynamic Equation 190 Points A 6 Kg Collar Travels Along Portoctly Smooth Horizontal Rod Defi 2 (36.24 KiB) Viewed 20 times
2. Prescribed Motion and Dynamic Equation (190 points). A 6 kg collar travels along portoctly smooth horizontal rod defined by the spiral equation r(t) = 2000, where is more in rodinns, as shown in Fig. 1. The motion of the collar in produced by an external controlled electromagnetic field such that the angular velocity is prescribed and constant with a value of 6(t) = Arad/s, for all t20. The initial condition is 0(0) = f rad. (a) Explain why in this case the sliding collar can be modeled as a particle (b) Using polar coordinates, i.e., {r(e),000)), right the position vector r(t) = r(t) (). (c) Using the Fundamental Theorem of Calculus (FTC) and initial condition 0(0), find 0(6), 20 18 16 FU Fun 14 12 10 8 r(t) r(t) = (2001) 6 6 4 2 0(t) 0 -10 8 op -6 -4 2 02 4 6 8 10 Figure 1: Prescribed Path
for t > 0. (d) Using the notion of time-derivative, find the angular acceleration õ(t), for £20. (e) Find the velocity of the collar in polar coordinates, ie, v(t) = dr(t) vy(t)ur(t) + vgus(t) dt = rur(t) + r(t)(b)us(t). (1) Find the acceleration of the collar in polar coordinates, i.e., dv(t) a(t) dt ar(t)u, (1) + aque(t) = [F(t) – r(t)º(c)] 4-(t) + [2+(!)ợct) +r(1)őtt)] no(t). (g) Using Newton's second law, find the instantaneous total force, Ftotal, acting on the collar as a function of time in polar coordinates, i.e., Flo(t) = F(t)ur(t) + Fo(t)uo(t) = mlay(t)ur(t) + agus(t)]. (h) Using Matlab, plot F-(t) over the range (0 : 2) s. (1) Using Matlab, plot Fo(1) over the range (0 : 2) s. 6) It can be shown that the instantaneous radius of curvature in polar coordinates is given by 12 p2 +2 (5) Find the instantaneous radius of curvature as a function of time. (k) Using your answer for (e), find the instantaneous speed of the collar (t) = ||v(t)lla + 2
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