PART III Problem Problem1 A computer dik tart rotating from ret at contant angular acceleration. If it take 0.750 to complete it econd revolution: a) How long doe it take to complete the firt complete revolution; b) What i the angular acceleration? ANSWER: a) 1.81 ; b) α = 3.84rad i 2 Problem 2 A machinit turn the power on to a grinding wheel, at ret, at time t = 0. The wheel accelerate uniformly for 10 and reache the operating angular velocity of 42 rad/. The wheel i run at that angular velocity for 39 and then power i hut off. The wheel decelerate uniformly at 2.6 rad/ 2 until the it top. In thi ituation, the angular acceleration of the wheel between t = 0 and t = 10 i cloet to: a) 5.9 rad/ 2 b) 7.6 rad/ 2 c) 4.2 rad/ 2 d) 5.0 rad/ 2 e) 6.7 rad/ 2 Partial Solution: Ue ω z = ω 0z + α z t α z = ω ω z 0z t with ω 0z = 0 and ω z = 42 rad Anwer: c Problem 3 A light triangular plate OAB i in a horizontal plane. Three force, F1 = 3 N, F2 = 1 N, and F3 = 9 N, act on the plate, which i pivoted about a vertical axe through point O. In Figure below, the magnitude of the torque due to force F1 about the axi through point O i cloet to: a) 1.1 N m b) 1.4 N m c) 0.90 N m d) 1.8 N m e) 1.6 N m Partial Solution: Here ue τ = r F that give τ 1 = rf 1 in60 0, where you hould be able to deduce that the angle between r and F 1 i 60 anwer e) Problem 4*** In the diagram on the next page, the ma i releaed from ret. A it decend the rope rotate the pulley and the hollow phere without lipping. T1 i the tenion in the bottom part of the rope, and T2 i the tenion in the top part of the rope, with T 1 T 2. i) Draw a free-body diagram on m, and ue Newton econd law (equation 5-1)to obtain, the equation,t 1 mg = ma, where a i the acceleration. ii) Draw a free-body diagram on the pulley that include the four force acting on it, and ue Newton econd law (equation 10-45) for rotation to obtain the equation T 1 r T 2 r = Iα p = ( I /r 2 )α p r.
iii) iv) Draw a free-body diagram on the hollow that include the three force acting on it, and ue Newton econd law (equation 10-45) for rotation to obtain the equation T 2 R = I α = ( 2/3)MR 2 α. Ue the no-lip condition, a = rα p and a = Rα, to find the acceleration of m, and the angular acceleration of the phere and pulley, a well a the tenion T1 and T2. Compare your anwer with the one you obtain by the energy method. ANSWER: a = 1.2mi 2 ;T1 = 36.9N and T2 = 8.8N. Hanging Box m = 4.3 kg Pulley,, r = 0.099 m Hollow Sphere,, M = 11 kg, R =0.820 Problem 5 A block of ma 5.00 kg i releaed from ret, lide down a urface inclined at 36.9 to the horizontal. The coefficient of kinetic friction i µ k = 0. 1. A tring attached to the block i wrapped around a olid cylindrical flywheel of M = 25.0 kg and R = 0.15 m. There i friction between the rope and the flywheel o that the rope doe not lip againt the wheel. L Solid cylinder with moment of inertia 36.9 L = 1.1 m A) Draw a free energy diagram on the block howing all force acting on it, and hence determine the kinetic force of friction on the block. B) Uing the conervation of total energy or the work-energy theorem, determine the linear peed of the block after it ha lid 1.1 m down the incline (a hown in diagram). Don t forget that the flywheel will be pinning.c) Calculate the angular velocity of the flywheel after the block ha lid 1.1m? ANSWER: A) f k = 3.92N ; B) Ue conervation of energy W ext = f k d = ΔE mech = ΔU + ΔK = mgh+ ( 1/2)mv 2 + ( 1/2)Iω 2, d = 1.1 m, h = din36.9, m = 5 kg, I = ( 1/2)MR 2, M = 25 kg, R =.15 m, with no-lip condition ωr = v, v = 1.1mi 1 ; C) ω = 7.441 1.
Problem 6 *** In the diagram below, a uniform pherical boulder tart from ret and roll down a 50.0 m high hill. The top half of the hill i rough o that the boulder roll without lipping, but the bottom half i covered with ice, and o i frictionle. What i the tranlational peed of the boulder when it reache the bottom of the hill? ANSWER: 29 m/ Point 1 25.0 m 25.0 m Rough Roll without lipping Point 2 Smooth Sphere lip Solid phere (table 9.2) + Clockwie M ma of phere i poitive R radiu of phere Point 3 Problem 7 In diagram below a 2kg rock i at point P traveling horizontally with a peed of 12 m/. At thi intant what i the magnitude and direction of the angular momentum? If the only force acting on the rock i it weight, what i the rate of change (magnitude and direction) of the angular momentum? Direction perpendicular to x-y plane indicate +z out of the page indicate z into page Alo +x right +y up Angular Momentum L = r p = m r v valid for point particle w.r.t. point O r, r = 8m Uing the right hand rule on it i eay to ee that the 36.9 143.1 v direction of i z or into the page For the magnitude L = L = mvr in143.1 " = 2kg 12m / 8m.6 L= 115.2kg-m 2 /.
Torque due to gravity on particle w.r.t. point O. τ = r F g,f g = mg = 2kg 9.8m / 2 = 19.6N r, r = 8m Uing the right hand rule on 53.1 direction of i +z or out the page. it i eay to ee that the 36.9 Thi can be expreed in a different unit Uing econd law for rotation in term of angular momentum. Hence the net torque i the rate of change of angular F g momentum Since the rate of change of angular momentum τ = d L / dt ha oppoite direction (+z) compared to the direction of the current angular momentum L = r p = m r v (-z), the angular momentum i decreaing. Can you ee the imilarity with our much earlier dicuion on linear kinetic? FINAL COMMENT AND ADVICE: Angular momentum L and Torque τ depend on the origin (O). For example, L = 0 and τ = 0 may be zero for origin O, but nonzero L 0 and τ 0 in another origin O /. Study angular momentum problem and tatic equilibrium problem. Problem 8 A carouel ha a radiu of 3.0 m and a moment of inertia of I C = 8000kg m 2, for rotation about axi perpendicular to the it center. The carouel i rotating unpowered and without friction with an angular velocity of 1.2 rad/. An 80-kg man run with a velocity of 5.0 m/, on a line tangent to the rim of the carouel, overtaking it. The man run onto the carouel and grab hold of a pole on the rim. +y +x Direction perpendicular to x-y plane indicate +z out of the page indicate z into page USE THIS CONVENTION TO INDICATE THE DIRECTION OF ANGULAR MOMENTUM a) Before the colliion, what i the magnitude of the angular momentum of the rotating carouel, L C, with repect to the center of the carouel? What i the direction of L kg i m2 C? ANSWER: L C = 9600, direction out of page, +z, or L C = 9600 kgim2 ˆk b) Before the colliion, what i the magnitude of the angular moment of the running 80-kg man, L M, with repect to the center of the carouel?
What i the direction of L kg i m2 M? ANSWER: L M = 1200, direction out of page (+z), or L C = 1200 kgim2 ˆk. c) After the colliion when the man i on the carouel, what i the magnitude of the final angular velocity of the carouel (with the man on it), ω fc? What i the direction of the final angular velocity ω fc? Note: I total = I C + mr 2. ANSWER: ω fc = 1.24 rad out of page or +z, ω fc = 1.24 rad ˆk