- 2 Prescribed Motion And Dynamic Equation 190 Points A 6 Kg Collar Travels Along A Perfectly Smooth Horizontal Rod De 1 (40.27 KiB) Viewed 35 times
2. Prescribed Motion and Dynamic Equation (190 points). A 6-kg collar travels along a perfectly smooth horizontal rod defined by the spiral equation r(t) = (2016), where 6 is measured in radians, as shown in Fig. 1. The motion of the collar is produced by an external controlled electromagnetic field such that the angular velocity is prescribed and constant with a value of h(t) = 4 rad/s, for all t> 0. The initial condition is e(0) = { rad.
20 18 16 Fut Frun 14 12 10 CO 8 r(t) r(t) = (20(t) = 6 4 2 e(t) Ur UO 0 -10 -8 -6 -4 -2 0 2 4 8 10 Figure 1: Prescribed Path.
(a) Explain why in this case the sliding collar can be modeled as a particle. (b) Using polar coordinates, i.e., {r(t), 8(t)}, right the position vector r(t) = r(t)ur(t). (c) Using the Fundamental Theorem of Calculus (FTC) and initial condition 0(0), find 0(t),
(d) Using the notion of time-derivative, find the angular acceleration ä(t), for t > 0. (e) Find the velocity of the collar in polar coordinates, i.e., dr(t) dt = v(t)ur(t) + vyuo(t) = rur(t) +r(t){t}ug(t). v(t) = (f) Find the acceleration of the collar in polar coordinates, i.e., a(t) dv(t) = a(t)u(t) + apuolt) dt = [F(t) – r(t)ô-(e)] 4-(t) + {2(t)ö(t) +r(t)őce)] _o(t). = (g) Using Newton's second law, find the instantaneous total force, F total, acting on the collar as a function of time in polar coordinates, i.e., Ftotal(t) = Fr(t)ur(t) + Fy(t)uo(t) = m [ar(t)u(t) + aguo(t)]. : (h) Using Matlab, plot Fr(t) over the range (0 : 2) s. (i) Using Matlab, plot Fo(t) over the range (0 : 2) s. 6) It can be shown that the instantaneous radius of curvature in polar coordinates is given by [x2 +($) ?] p= p2+2 () Find the instantaneous radius of curvature as a function of time. -T. (k) Using your answer for (e), find the instantaneous speed of the collar v(t) = ||v(t)|2 (1) Using the formula 를 s(0) = -5° (10) 1 + Carm) de de, find the instantaneous distance function as a function of time, s(t), for t > 0.
(m) Using your answer for (1), find the instantaneous speed of the collar by computing ds(t) v(t) = dt (n) Are your answers for (k) and (m) consistent? Explain and discuss. (0) Write the velocity of the collar in tangential and normal coordinates, v(t) = v(t)u:(t). (p) Write the acceleration of the collar in tangential and normal coordinates, a(t) = at(t)ut(t) + an(t)un(t) va(t) = i(t)u(t) + plt) un(t). (q) Using Newton's second law, find the instantaneous total force, Ftotal, acting on the collar as a function of time in tangential and normal coordinates, i.e., Ftotal(t) = Ft(t)u:(t) + Fn(t)un(t) = m (at(t)ut(t) + an un(t)). (r) Using Matlab, plot F(t) over the range (0 : 2) s. (s) Using Matlab, plot Fn(t) over the range (0 : 2) s.