chap_IV_hwk_fall_2010

chap_IV_hwk_fall_2010 - Particle K inetics Homework...

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Unformatted text preview: Particle K inetics Homework Problems ME 274 IV-6 Chapter IV: Particle Kinetics Homework Problem IV-4 Problem IV.4 The mechanism, shown below, consisting of a rotating arm, frictionless slot, and block of mas m, i driven at a shown below, ! by a motor rotating arm, frictionless slot, and Given:s The smechanism, constant speedconsisting of a attached at point O. Determine: block of mass m,a) tdriven at a in the cable and ω by a motor exerted on the block at the instant shown ( is he tension constant speed normal force attached at point O. corresponding to ! = 10 rad/s Find: Determine: (a) the tension in the cable and normal force exerted on the block at the instant shown (b) the minimum ω = 10rad/sec, andnecessary to keep angular velocity ω necessary to keep corresponding to angular velocity ! b) the minimum the cable taut at this instant the cable taut at this instant. ! r g m O " Use the following parameters in your analysis: r = 0.3m, m = 12kg and θ = 60◦ . Use r = 0.20 m, m = 10 kg and " = 53.13° in your analysis. Kinetics Homework Problems ME 274 IV-8 m IV-6 Chapter IV: Particle Kinetics Homework Problem IV.6 um of an industrial polisher, shown an industrial polisher, point Obelow,arotates about point O with a constant Given: The drum of below, rotates about shown with constant !. A small cube of ω . A small cube of aluminuminner surfacethe inner surface of the drum. It is observed that speed aluminum is placed on the is placed on of the drum. It is ed that this piece of metalmetalno motion relative to the the polishers surface as it passes through θ = 0. this piece of has has no motion relative to polisher’s surface as it through " = 0. Determine the relationship among r, µS (the static coefficient of Find: Determine the relationship among r, ( S (the static coefficient between the block and the drum’s inner surface) and "crit µthe angle of orientation of friction between the block and the h the block first slips).drums inner surface) and θcritical (the angle of orientation at which the block first slips). ! g µs O r " m etics Homework Problems IV-10 IV-8 ME 274 Chapter IV: Particle Kinetics Homework Problem IV.8 Given: The collar, from below, of mass A starts from rest is r, shown below, of mass m, starts shown rest at point A. m, constant force F at point A. A constant force F is the collar in applied to the shown.in the direction shown.of the that the mechanism lies in the vertical plane. the direction collar Determine the speed Note collar when it Assumemechanism liesbe smooth. oint B. Note that the all surfaces to in the vertical plane. Assume all surfaces oth. Find: Determine the speed of the collar when it reaches point B. A m F r g B ollowing: mg =Useb, F = 30 lb and r = 1.5 ft.in your analysis: mg = 10lb, 50lb and r = 2f t. 6 l the following parameters Chapter IV: Particle Kinetics Homework inetics Homework Problems m IV-9 ME 274 IV-11 Problem IV.9 Given: The collar shown below, of mass m is pushed against a spring with stiffness k until it is llar, shown below, of mass mandpushed against a spring fficient of kineticuntil it isbetween the collar and rod is compressed ∆ is then released. The coe with stiffness k friction ssed !, and then. released. Assuming that there is a kinetic coefficient of friction µk een the collar and rod, and that the collar fails to launch clear of the rod, Find: Determine the maximum height measured the the as measured from the point of release. ine the maximum height obtained by the collar, as obtained byfromcollarpoint of Assume that the collar remains on the rod at all times. . g m µk ! k following: m Use.5 kg, k = 2000 N/m, ! = 0.3your analysis: and !4= 36.87°. N/m, µk = 0.15 and θ = 30◦ . = 2 the following parameters in m, µk = 0.20, m = kg , k = 2500 Problem IV-11 Pellets A and B (having masses m and 3m, respectively) are placed within a smooth tube trapping a small compartment of fuel. At a time when the pellets are initially at rest, the Chapter ignited. The Kinetics Homework over a short time !t, and over this time the IV-13 fuel is IV: Particle combustion occurs combustion applies equal and opposite forces on the pellets with this force idealized by the force time h Problem IV.11 istory F(t) shown below. a) Draw individual free body diagrams (FBD’s) for pellets A and B. From these Given: Pellets Adetermine the speed of each pellet at a time t , where t > !within a smooth tube FBD’s, and B (having masses m and 3m, respectively) are placed t. 2 2 trapping a small compartment of fuel. At a time when the pellets are initially at rest, the fuel is b) Draw an FBD for pellets A a short time ∆t, and over and B as the combustion ignited. The combustion occurs over and B together (i.e., treat A this time a single system).applies F opposite forces on the pellets with this force momentum the force time history F(t) equal and rom this FBD, provide an argument that linear idealized by for the system of A and shown below. B together should be conserved. Use your results from a) above to verify this argument. Find: Determine the speed of each pellet at time t2 , where t2 > ∆t. F(t) A B m 3m smooth F0 smooth combusting fuel 0 !t Use the following parameters in your analysis: F0 = 3000N , ∆t = 0.005sec and m = 0.75kg . Use the following parameters in your analysis: F0 = 2000 N , ! t = 0.006 s and m = 0.5 kg . t particle impacts the floor with a speed of v1. This impact of the particle with the floor lasts for a short duration of time !t, and after the impact is complete, the particle rebounds upward with a speed of v2. The particle continues upward reaching a maximum IV-14 height of h2. Chapter IV: Particle Kinetics Homework a) Draw a free body diagram (FBD) of the particle for the time period during its impact with the floor. Problem IV.12 b) Based on your FBD in a), determine the average force acting on the particle by Given: A particle of mass m is dropped from rest when at avalue ofha force Fa rigid floor. The the floor during impact. Recall that the average height 1 above (t) acting over a particle impacts the floor with a speed of v1 . This impact of the particle with the floor lasts for a !t short duration period of time !t after the by: F is= 1 of time ∆t, and is given impact complete,( t ) dtparticle rebounds upward with a F the . ave ! tmaximum height of h2 . speed of v2 . The particle continues upward reaching a 0 ! Find: For c) Rproblem: a)cdetermine the b) aboveforcenow ignorethe particle force during during this epeat your alculations in average but acting on the weight by the floor the time of of gravity, b) determine for the average acting on the particle by the impact in the presenceimpact. Did your answerthe average forceimpact force change much? floor during impact Determine the of gravity, c)/ compare your answers from a) and b), and d) determine d) in the absence value of h2 h1 . the value of h2 /h1 . v=0 v=0 h1 h2 v = v2 v = v1 Use the following parameters in your analysis: ∆t = 0.002sec, m = 15kg , v1 = 80m/sec and Use the v2 = 50m/sec. following parameters in your analysis: ! t = 0.001 s , m = 10 kg , v1 = 60 meters / sec and v2 = 45 meters / sec . Particle K inetics Homework Problems ME 274 IV-16 Chapter IV: Particle Kinetics Homework Problem IV-14 A cannonball P of mass m is fired toward a steel barrier on a stationary cart. At some Problem IV.14 time after rebounding from the barrier, the cannonball is observed to have a speed of vP a A moving in a of mass m is fired toward the figure. Let M stationary cart. mass of time Given: nd cannonball P direction shown below in a steel barrier on abe the combined At some the cannon and cart. Assume that the cart is able to move to have a speed of vP the after rebounding from the barrier, the cannonball is observed without friction along and moving in horizontal below and ignore the influence of combined mass a direction shown surfacein the figure. Let M be the air resistance. of the cannon/cart. Assume that the cart is able to move without friction along the horizontal surface and ignore the influence a) D of air resistance.etermine the velocity (both magnitude AND direction) of the cart after the cannonball bounces off the steel barrier at the instant shown below. b) Lproblem: a) determine the velocity vector of the cartof the cannonball and bounces Find: For this et !t represents the elapsed time between the firing after the cannonball the instant shown below. Determine the average value of the elapsed time between off the steel barrier at the instant shown below, and b) if ∆t representsthe horizontal force actingthe on the combined cannon and cart over the time period of 0 < t < ! of firing of the cannonball and the instant shown below, determine the average value t. the horizontal force acting on the combined cannon/cart over the time period of 0 < t < ∆t. vP P ! m cannon cart M Use the following parameters in your analysis: mg = 60lb, M g = 200lb, ∆t = 0.2sec, θ = 20◦ and vP = 120f t/sec. Use the following parameters in your analysis: mg = 50 lb , Mg = 250 lb , ! t = 0.4 s , ! = 25° and vP = 100 ft / s . Particle K inetics Homework Problems ME 274 Chapter IV: Particle Kinetics Homework IV-17 Problem IV-15 A particle of mass m is projected horizontally to the right at a height h above a smooth, Problem IV.15 horizontal floor with a speed of v0. The particle strikes the floor at a horizontal distance d from where it was is projected horizontally to the right restitution h above a smooth, Given: A particle of mass m initially projected. The coefficient of at a heightof the impact of the particle with the floor is The particle strikes the floor at a horizontal distance d from horizontal floor with a speed of v0 . e. where it was initially projected. The coefficient of restitution of the impact of the particle with the a) Determine the angle !1 that the velocity of the particle has with the horizontal floor is e. before impact. Use conservation of energy prior to impact. Find: For thisbproblem: a) determine thethat the 1 that the velocity of the particle has with the ) Determine the angle !2 angle θvelocity of the particle has with the horizontal horizontal immediately before impact,impact. determine the angle θ2 that the mentum of the particle immediately after and b) Use conservation of linear mo velocity in the horizontal has with the horizontal immediately after impact. direction along with the coefficient of restitution equation during impact. v0 v2 h !2 d BEFORE impact !1 v1 immediately AFTER impact Use the following parameters in your analysis: = f t , h v125 ft a t/sec Use the following parameters in your analysis: e = 0.5, h =e1500.6 and = 0 = 160fnd v0. = 150 ft / sec . A satellite P of mass m is in orbit around a planet whose center is at E. At position 1 shown below, the satellite is at a distance R1 from E and moving with a speed of v1. At position 2 shown, P is at a distance of R2 from E and moving with a speed of v2 with the Chapter IV: Particle Kinetics Homework velocity of P oriented as shown below. Assume that the only force acting on the satellite is a gravitational force directed toward E, and that the planet is not accelerating. IV-19 Problem IV.17 a) Determine the angular momentum of P about E at positions 1 and 2. Given: A satellite P of mass m is in orbit aroundthe planet whose center about E. is position 1 b) Provide an argument to support the claim that a angular momentum is at E At shown onstant.the satellite is at a distance R1 from E and moving with a speed of v1 . At position c below, 2 shown, P is at a distance of R2 from E and moving with a speed of v2 with the velocity of P c) Based on b) below. Assume that speed of P at acting on v2 in terms of gravitational force oriented as shown above, determine the the only force position 2,the, satellite is aR1, R2, directedatoward E, and that the planet is stationary. !2 nd v1. d) Explain why your result in c) does not depend on the gravitational force acting on Find: For this problem: a) determine the speed of P at position 2, v2 , in terms of R1 , R2 , θ2 and v1 , the satellite. and b) provide an argument explaining why your result in a) does not depend on the gravitational force acting on the satellite. e! eR !2 v2 e! P R2 E R1 v1 P eR Problem IV-20 Particles A and B each have a mass of m. A is constrained to move on a horizontal arm OC, and B is constrained to move on the vertical shaft about which arm OC rotates. A taut cable connects A and B as shown in the figure below. At an insIV: Particle Kinetics Homework IV-22 Chapter tant when the shaft is rotating with a rate of ! = !1 and when R = R1 , A and B are released from rest relative to the rotating arm and shaft. Consider all surfaces to be smooth, assume the masses of the Problem IV.20 arm, shaft and pulley to be negligible, and assume the radius of the pulley to be small. Given: Particles A and B each have a mass of m. A is constrained to move on a horizontal arm OC, and B is constrained to move on the vertical shaft about which arm OC rotates. A taut cable connects A and the angular speed of the arm ! 2 . At an instant when angular is rotating with a a) determine B as shown in the figure below. Consider using the the shaft rate of omentum equation. R1 , A and B are released from rest relative to the rotating arm and mω = ω1 when R = shaft. Consider all surfaces to be smooth, assume the masses of the arm, shaft and pulley to be b) determine the speed radius of the pulley the work-energy equation along with negligible, and assume the of B. Consider usingto be small. When A has moved outward to a position of R = R2 , appropriate kinematics relating the motions of A and B. Be careful – the velocity Find:of A has two components at position 2 (one due = R2 : a) determine the angular speed of the When A has moved outward to a position of R to the rotation of the arm and one due to its sliding along particle B. arm ω2 , and b) the speed of the arm). R C O A g B ! Use the following parameters in your analysis: m = 15kg , R1 = 0.3m, R2 = 0.6m and ω1 = Use rad/sec. the following parameters in your analysis: m = 10 kg , R1 = 0.2 meters , 25 R2 = 0.5 meters and !1 = 20 rad / sec . Chapter IV: Particle Kinetics Homework IV-23 Problem IV.21 tics Homework Problems Given: The system shown is released from rest. ME 274 IV-21 (3/27) Find: On release, determine: a) the acceleration of each block, and b) the tension in the cable. shown is released from rest. the following: acceleration of each block. tension in the cable. ! Use the following parameters in your analysis: µs = 0.25, µk = 0.2 and θ = 30◦ . llowing parameters in your analysis: µ s = 0.25 , µ k = 0.20 and ! = 30° . Chapter IV: Particle Kinetics Homework IV-31 Problem IV.29 Given: A bullet strikes the lower particle of a stationary pendulum with a speed of v. After impact, the bullet sticks to the particle. mework Problems 9 (3/248) ME 274 Find: Determine: a) the angular speed of the pendulum immediately after impact, and b) the maximum rotation angle through which the pendulum swings after impact. M the lower particle of a stationary a speed of v. After impact, the bullet ticle. Determine: lar speed of the pendulum immediately pact. imum rotation angle through which the m swings after impact. M m Use the following parameters in your analysis: v = 300m/sec, m = 50grams and M = 3.2kg . ng parameters in your analysis: v = 300 m / sec , m = 50 grams and Particle K inetics Homework Problems M Chapter IV: Particle Kinetics Homework -31 Problem IV IV-33 A basketball is being thrown upward with an initial velocity of vi. Using Newton’s second law, determine the maximum height of the basketball Problem IV.31 a) when drag force caused by air resistance is neglected. 2 b) w upward force magnitude is given by Given: A basketball is being thrown hen dragwith an initial velocity of vi . Fdrag = cv . Find: Determine the maximum height of the basketball: a) in your analysis:caused .62 air resistance Use the following parameters when drag force m = 0 by kg, vi = 10 m/s, c = 0.013 N 22 is neglected, and b) when drag force magnitude is given by Fdrag = cv 3 . sec /m h g vi Use the following parameters in your analysis: m = 0.7kg , vi = 10m/sec and c = 0.015N − sec2 /m2 . Chapter IV: ParticleParticle KineHomework Problems Kinetics tics Homework IV-35 Problem IV.33 ME 274 Problem IV-33 ! A mass is a spring rotating about the about O. Given: A mass is attached to attached to a spring rotating pin at the pin at O. At the instant when ! = 1 rad/s 2 ! r = constant, r = 0.8 m, r = 2 m/s and !! = 2.4 m/s . Find the unstretched length of the spring. (m = 2 length=of 0 N/m) kg, k 8 the spring. Find: Determine the unstretched O ! g r m Use the following parameters in your analysis: m = 3kg , k = 100N/m, r = 0.75m, r = 3m/sec and ˙ r = 3m/sec2 . ¨ Particle K inetics Homework Problems ME 274 Problem IV-44 IV-46 Chapter IV: Particle Kinetics Homework A thin homogeneous bar OA (having a mass of M = 10 kg and length L = 3 meters) is pinned to ground at point O in such a way that the bar moves in a HORIZONTAL plane. Problem IV.44 Particle B (with a mass of 10 kg) is able to slide without friction on OA, and is attached to A with a spring having a stiffness of k = 500 N/m and an unstretched length of 1.5 Given: A thinthe instant shown, bar OA is rotating CW M and speed of ) is pinned to ground at meters. At homogeneous bar OA (having a mass of with a length L , point O in such a way that the bar moves in a HORIZONTAL plane. Particle B (with a mass of particle B is not moving relative to bar OA and the spring is unstretched. m) is able to slide without friction on OA, and is attached to A with a spring having a stiffness of k andind the velocity of particle d. after it has moved 0.5 meters outward on the bar. Write F an unstretched length of B At the instant shown, bar OA is rotating CW with a speed of ω1 , particle B is notamoving relative to bar OA and the spring is unstretched. your answer as vector. H Determine thebar OA and particle B as a single sytstem. a distance of b meters outward on Treat velocity of particle B after it has moved Find:INT: the bar. Write your answer as a vector. O M (mass of bar) !1 B L m (mass of B) (B can slide on rod) k A HORIZONTAL PLANE Use the following parameters in your analysis: ω1 = 9rad/sec, m = 10kg , M = 10kg , L = 3m, k = 500N/m, d = 1.5m and b = 0.5m. Particle K inetics Homework Problems ME 274 Problem IV-45 Chapter IV: Particle Kinetics Homework masses of mA = mB = 10 kg) are interconnected by IV-47 Given: Particles A and B (having the cable-pulley system shown in the figure. Both particles are constrained to vertical motion with particle A able to slide on a smooth vertical rod. The Problem IV.45 system is released at sA = 0 with A traveling downward with a speed of 5 m A and B (having masses of m small, massless and frictionless. Given: Particles /sec. Assume the pulleys to beA and mB ) are interconnected by the cable-pulley system shown in the figure. Both particles are constrained to vertical motion with particle A able to slide on a smoothare asked rod. The system peed of particleA = 0 with A travelingthe Find: You vertical here to find the s is released at s A when A has reached downward with a speed of position of s the pulleys to be small, massless and frictionless. vA1 . Assume = 2 m. A Find: Find the speed of particle A when A has reached the position of sA . 1.5 m O sA g O sB A B A B smooth rod Use the following parameters in your analysis: mA = 10kg , mB = 10kg , vA1 = 5m/sec and sA = 2m. s A and B (having masses of m and M, respectively) are initially IV-48 Chapter IV: Particle Kinetics Homework ing in directions perpendicular to each other with speeds of vA1 and vB1 , tively, as shown below in the figure. After impacting each other, A is ing to the RIGHT with a speed of vA 2 , and B travels with a speed of vB 2 . Problem IV.46 der all surfaces to be smooth. Given: Blocks A and B (having masses of m and M , respectively) are initially traveling in directions perpendicular to each other with speeds of vA1 and vB 1 , respectively, as shown below in the is problem, figure. After impacting each other, A is traveling to the RIGHT with a speed of vA2 , and B travels ) draw FBD’swith A speed of vB 2 (direction of Imotion a set of coordinate is not known). Consider all surfaces to for a alone, B alone and A+B. nclude for B after impact axes with thesesmooth. be FBD’s. ) determine the mass M of block B. Find: For this problem: a) determine the mass M of block B, and b) determine the coefficient of ) determine the coefficient, of restitution, e,of A the impact of A and B. restitution, e for the impact for and B write down the linear momentum vector ns for A, B and A+B in terms of their and tangential components, as well as ficient of restitution equation. Use these ns to find M and e. B A vA1 m g parameters in your analysis: m = 3kg , eters / sec , vA 2 = 2 meters / sec rs / sec . M vB1 HORIZONTAL PLANE Use the following parameters in your analysis: m = 3kg , vA1 = 4m/sec, vB 1 = 4m/sec, vA2 = 2m/sec and vB 2 = 5m/sec. Problem IV-50 Given: A smooth, circular slot is cut into block A with block A being constrained to move along a smooth, horizontal surface. The slot is vertical at the top surface of the block with the slot being horizontal at IV: Particle Kinetics Homework IV-52 Chapter the right edge of the block, as shown in the figures below. In Position 1, block A is stationary, and a particle B is released from rest into the upper opening of the slot. At Position 2 shown Problem IV.50 below, particle B is exiting the slot at the right edge of the block. The masses of A and B are M = 40 kg and m = 20 kg, respectively. The radius of the Given: A smooth, circular slot is is given by r = 2 meters. circular slot cut into block A with block A being constrained to move along a smooth, horizontal surface. The slot is vertical at the top surface of the block with the slot being horizontal at the right edge of the block, as shown in the figures below. In Position 1, block A is Find Determine the velocity of block the upper elocity of the slot. at Position 2 stationary, and :a particle B is released from rest into A and the vopeningof particle B At position 2. shown below, particle B is exiting the slot at the right edge of the block. The masses of A and B You must provide an radius of the circular slot is r are M and m, respectively. Theaccurate free body diagram of .the system used in your solution in order to receive full credit for your work. Consider using both the linear impulsemomentum and work- block equations in solving particle B at Find: Determine the velocity ofenergyA and the velocity of this problem. position 2. m VA2 B r g B A M POSITION 1 A POSITION 2 Use the following parameters in your analysis: m = 20kg , M = 40kg and r = 2m. VB2 oblem IV-52 ven: Particle P (of mass m = 8 kg) moves within a vertical plane inside a rough, circular slot. The coefficient of kinetic friction between particleChapter IV: Particle Kinetics Homework P and the slot IV-54 is µk = 0.2, and the radius of the slot is r = 2 meters. At the position shown below, the speed of P is known to be v = 3 m/sec. Problem IV.52 d: For this position: Given: Particle P (of mass m) moves within a vertical plane inside a rough, circular slot. The a) coeetermine the numerical value of the normalPcontact force of the slot the radius of the slot is r. D fficient of kinetic friction between particle and the slot is µk , and Atn P. position shown below, the speed of P is known to be v . o the b) Find:in contact position: ia) determine the numerical value of the normal contact force of the slot Is P For this with the nner or outer surface of the slot? c) on etermine the rate of change of speed of of speed of P. D P, and b) determine the rate of change P. v rough surfaces, m P µk r 53.13° O g Use the following parameters in your analysis: m = 8kg , µk = 0.2, r = 2m and v = 3m/sec. Chapter IV: Particle Kinetics Homework IV-55 Problem IV.53 Given: Blocks A and B (having masses of 2m and m, respectively) are connected by rigid, massless rod AB of length L. Block A is constrained to move along a smooth vertical guide, and B moves along a smooth, horizontal surface. The system is released from rest when A is at a height of h1 Homework Problems ME 274 ABOVE the path of B. -53 Find: cks A and B (having Determine 2m and m, of block A when A has dropped to a position that is at a distance of masses of the speed respectively) are connected by h BELOW the path of B. id, massless rod 2AB of length L. Block A is constrained to move along a ooth vertical guide, and B moves ng a smooth, horizontal surface. e system is released from rest when 2m s at a height of h1 ABOVE the path A B. r this problem, g ) Draw a free body diagram of the system made up of block A, block B and rod AB. m L h1 B ) Determine the speed of block A when A has dropped to a position that is at a distance of h2 BELOW the path of B. ysis, use the following parameter 10 kg, k = 1000 N/m, L = 0.5 0.4 meters and h2 = 0.3 meters. Use the following parameters in your analysis: m = 10kg , k = 1000N/m, L = 0.5m, h1 = 0.4m and h2 = 0.3m. ...
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