Initials: Problem 1 - 25 points A Bobsled challenge requires that the competitors clear a vertical circular loop of radius R, after starting at the same height as the top of the loop. You may assume that the entire track is frictionless all the way through the end of the loop, and that the riders have negligible mass compared to the bobsled of mass m. The acceleration due to gravity has magnitude g. starting point in part (c) starting point in part (a) loose snow a- Explain whether or not the bobsled team can make it through the loop - meaning remain in contact with the track at all times - if it starts from rest at the end of launching section. b- Determine the minimum speed required at the top of the loop to ensure that the bobsled remains barely in contact with the track. Hint: a free body diagram at the top of the loop might be helpful.c- Determine the minimum constant force the back rider needs to apply from at Initials throughout the horizontal launching section of length L in order to make it through the loop. Hint: think about the minimum speed required at the end of the launching section. d- Determine the minimum distance D that is required for the bobsled to stop after the loop, assuming a coefficient of kinetic friction / on loose snow, if the bobsled barely makes it through the loop. Hint: think about the bobsled speed as it exits the loop.Initials: Physics 8A MT2 Equation Sheet r (t ) = x(t)ity(t)j Ar = 12- r1 = Axi+ Ayj Newton's Ist Law AT B = constant = ) F = o At VAV = = VAV-xi + VAV-yj Newton's 2nd Law D = lim = = Vxi + Vyj 4t-0 4t F = ma av aAV = At -= aAV-xi + aAV-y] Newton's 3rd Law a = lim = axi + ayj OFMe/You = -Fyou/Me v (t ) = vo + alt - to] Tension: T r(t) = ro+ volt- tol + alt - to] Ax = X2 - X1 Ax _X2 - X1 Normal Force VAV-x At t2 - t1 Ax dx Vx = li At-o At dt Avx Vx2 - Vx1 Gravity aAV-x At t2 - 1 Fg = mg AVX - dvx in (904) ax = lim - dt Static Friction Vx (t) = Vox + ax[t - to] 1 fs SUSN x(t) = xo + Vox[t - tol+ = ax[t - to]2 Kinetic Friction v2 = Vox + 2ax[x - xo] fK = HRN Ay = y2 - V1 Ay _y2 - y1 VAD - y Spring Force At t2 - t1 Ay dy Fsp = - kx (x measured from relaxed position) Vy = lim - At-0 At dt Avy Vy2 - Vy1 Uniform Circular Motion aAV-y At t2 - t1 Avy dvy a = ay = lim - At- 0 At dt v = rw Vy (t) = Voy + dy[t - to] T = - y(t) = yo + Voy[t - to] + 5 ay[t - to]2 vz = vay + 2ayly - yo]Initials: Work and Energy amounts B [ Fext = 0 = AP = 0 W = F . dr m1 - mz 2m2 V1f = 2 vjit- - Vzi my + mz mi + mz W = FL cosOF (constant angle and magnitude) 2m1 Vli +- m2 - m1, V2f = - Vzi Wg = -mg4h (height axis pointing upward) mit m2 mi + mz WSp = -(x3 - XA) (x measured from relaxed position) Rotational Kinematics 1 K = =mv2 0(t) = angular position 40 = 02 - 01 Wnet = > Wi = 4K 40 02 - 01 WAV = At t2 - t1 U + K =E de AE = WNC and AU = -Wc w = lim At-0 At at Ug = mgh (height axis pointing upward) AW W2 - W 1 a AV Usp = = kx2 (x measured from relaxed position) At t2 - t1 Aw dw a = lim - At-0 4t dt Universal Gravitation w (t ) = wot alt - tol FG = GMm 0 (t) = 00 + wolt - to] +5 alt- to]? GM g = - 7 = wg + 2a[0 -Oo] GMm s = ro (arc length) UG = = rw atan = ra GM 2 2 Vorb = arad = - =rw2 Rotational Dynamics Mys 2173/2 T =- VGM I = Et=1mir? (point masses) Ip = ICM + Md2 (parallel axis theorem) 2GM It| = rF sino notVesc = [Text = 1a Linear Momentum p = mi P = > Pi - Quadratic Equation [ Fest = at ax2 + bx + c = 0 has the solutions X = 2a -(-bvb2 - 4ac)Derivatives d(x12) Initials: = nx -1 dx Integrals dx = X2 - X1 Lengths, areas and volumes Circumference of circle: 2nR Area of disk: TR Surface area of sphere: 4TR- Lateral area of cylinder: 2ntRh Volume of cylinder: ntR h Volume of sphere: 4TR3/3 Trigonometry sin (90) = cos (09) = - cos (1809) = 1 sin (0) = cos (90) = sin (180) = 0 cos (60) = sin (30) = 1/2 sin (60) = cos (309) = V3/2 sin (450) = cos (450) = 12/2 cos (180- 0) = - cose sin (180- 0) = sine cos (90 + 0) = - cos (90 - 0) = - sine sin (90 + 0) = sin (90 - 0) = cose cos' 0 + sin- 0 = 1Initials: Table of rotational inertias M is the mass of the system Axis Axis Axis Hoop about central axis Annular cylinder Solid cylinder (or ring) about (or disk) about central axis central axis 1=MR2 (a) 1 = *M(R, 2 + R2?) ( b ) 1 = 4MR (C) Axis Axis Axis Solid cylinder Thin rod about Solid spherea (or disk) about axis through center bout any central diameter perpendicular to diameter length 2P 1= +MR + + ML2 (P) 1= LML? (e) 1 = 3MR (D) Axis Axis Axis Thin Hoop about any Slab about spherical shella diameter perpendicular bout any axis through 2R diameter center 1 = 1MR2 (8) 1 =MR (h ) 1 = 12 M(al + b? ) (1)