Initials: Star 2 Problem 2 - 25 points 2M A spaceship finds itself located so that it is exactly at distance D from three stars, in 1/ the configuration shown in the diagram. Star 1 Star 3 Two of them (stars 1 and 3) have mass M, the middle one (star 2) has mass 2M, and M P M Ms the spaceship has mass ms. D D a- Determine the direction and magnitude of the gravitational force acting on the spaceship due to Star 1 alone. Use the (x,y) coordinate system defined in the diagram. b- Determine the direction and magnitude of the total gravitational force acting on the spaceship due to all three of the stars.c- Determine the gravitational potential energy of the spaceship due to all three stars. Hint: the gravitational potential energy of the sp Initials: potential energies from each star individually. the spaceship due to all three stars is the sum of the d- Determine the minimum speed at which the spaceship should be moving at its current location if it is to be able to travel infinitely far away from the stars without having to burn any more fuel. You may assume that the three stars do not move very far from their current configuration while the spaceship leaves the system.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)