Particle Kinetics Homework

Transcription

1 Chapter IV: Particle Kinetics Homework Chapter IV Particle Kinetics Homework IV-1

2 Chapter IV: Particle Kinetics Homework IV-2

3 Chapter IV: Particle Kinetics Homework inetics Homework Problems ME 274 Homework 4. m IV-1 Given: The truck shown below is travelling along I-65 when a deer runs out onto the highway. k shown below The is travelling truck is initially along I-65 traveling when at a deer a speed runs ofout v 0 onto and decelerates the highway. at aif constant rate. k is initially travelling at a speed of v 0 and decelerates at a constant rate, find the Find: Find the minimum distance s over which the truck can stop to ensure that its load does not m distance s over which the truck can stop to ensure that its load does not shift shift (and the deer is safely avoided). deer is safely avoided). following: v 0 Use = 30the m / following sec, W = parameters 1000 N and in your µ s = 0.4 analysis:. v 0 = 40 m/s, W = 1000 N and µ s = 0.2. IV-3

4 ticle Kinetics Homework Problems ME 274 oblem IV-52 ven: Particle P (of mass m = 8 kg) moves within a vertical plane inside a rough, Chapter IV: Particle Kinetics Homework circular slot. The coefficient of kinetic friction between particle P and the slot is µ k = 0.2, and the radius of the slot is r = 2 meters. t the position shown Homework 4. below, the speed of P is known to be v = 3 m/sec. Given: Particle P (of mass m) moves within a vertical plane inside a rough, circular slot. The d: For this coefficient position: of kinetic friction between particle P and the slot is µ k, and the radius of the slot is r. a) t Determine the position the shown numerical below, value theof speed the normal of P is contact known to force be v. of the slot on P. Find: For this position: b) (a) Is P Determine in contact the with numerical the inner value or outer of the surface normal of contact the slot? force of the slot on P; c) (b) Determine the the rate rate of change of change of speed of speed of of P. P. v P m r rough surfaces, µ k g Use the following parameters in your analysis: m = 8 kg, µ k = 0.2, r = 2 m and v = 3 m/s. IV-4

5 Chapter IV: Particle Kinetics Homework Homework 4.C Homework Problems ME 274 Given: The mechanism, shown below, consisting of a rotating disk, frictionless slot, cable, and -5 block of mass m, is driven at a constant speed ω in the horizontal plane by a motor attached at point. ism, shown below, consisting of a rotating disk, frictionless slot, cable, and ss m, is driven Find: at a Determine: constant speed! in the horizontal plane by a motor oint. Determine: (a) The tension in the cable; (b) The normal force exerted on the block. nsion in the cable ormal force exerted on the block! m h d r HRIZNTL PLNE Use the following parameters in your analysis: ω = 12 rad/s, h = 0.4 m, d = 0.6 m and m = 10 kg. wing:! = 10 rad / sec, h = 0.3 m, d = 0.4 m, and m = 4 kg. IV-5

6 Particle Kinetics Homework Problems ME 274 Chapter IV: Particle Kinetics Homework Homework Problem 4.D IV-4 Given: The mechanism, The mechanism, shown shown below, below, consisting consisting of a rotating of a rotating arm, arm, frictionless frictionless slot, slot, and block and block of of mass mass m, m, is driven is driven at aat a constant speed speed ω by! by a motor a motor attached at at point.. Determine: (a) the tension in the cable and normal force exerted on the block at the instant shown Find: Determine: (a) The tension corresponding in the cable to! at = the 10 rad/s instant shown provided ω = 10 rad/s; (b) The (b) normal the minimum force exerted angular onvelocity the block! provided necessary ω to = keep 10 rad/s; the cable taut at this instant (c) The minimum angular velocity ω necessary to keep the cable taut at the instant shown.! g r m " Use the following parameters in your analysis: r = 0.3 m, m = 12 kg and θ = 60. Use r = 0.20 m, m = 10 kg and " = in your analysis. IV-6

7 Kinetics Homework Problems ME 274 Chapter IV: Particle Kinetics Homework m IV-2 Homework 4.E stem shown below is released from rest in the configuration shown. If the masses Given: The system shown below is released from rest in the configuration shown. are 50 kg and 75 kg, respectively, µ k = 0.2, µ S = 0.3, and! = Determine: the acceleration Find: of block Determine: the acceleration (a) of The block acceleration of block ; (b) The acceleration of block ; the tension T in the cable (c) The tension in the cable. g m m! µ s, µ k Use the following parameters in your analysis: m = 40 kg, m = 80 kg, µ s = 0.3, µ k = 0.15 and θ = 60. IV-7

8 Problem IV-56 Given: Particle P of mass m slides in the direction shown within a tube with a speed of v rel relative to the tube as the tube rotates in the Chapter CCW IV: sense Particle with an Kinetics angular Homework speed of! and angular acceleration!. Homework 4.F Find: For this problem: Given: Particle P of mass m slides in the direction shown within a tube with a speed of v a) Determine the acceleration of P. rel relative to the tube as the tube rotates in the CCW sense with an angular speed of ω and angular acceleration α. b) Determine the friction force acting on P. Find: Use the For following this problem: parameters in your analysis: m = 10 kg,! = 5rad / sec, (a)! = Determine 2rad / sec the 2, acceleration v of P; rel = 4m / sec, µ s = 0.6, µ s = 0.3, L = 0.4meters and (b) h = Determine 0.2meters the. friction force acting on P. y µ k, µ s v rel!," P h x L HRIZNTL PLNE Use the following parameters in your analysis: m = 10 kg, ω = 5 rad/s, α = 2 rad/s 2, v rel = 4 m/s, µ s = 0.6, µ k = 0.3, L = 0.4 m and h = 0.2 m. IV-8

9 Particle Kinetics Homework Problems ME 274 Problem IV-45 Chapter Given: IV: Particles Kinetics and Homework (having masses of m = m = 10 kg) are interconnected by the cable-pulley system shown in the figure. oth particles are constrained to vertical motion with particle able to slide on a smooth vertical rod. The Homework 4.G system is released at s = 0 with traveling downward with a speed of 5 Given: Particles m/sec. and ssume (having the pulleys masses to of be m small, and mmassless ) are interconnected and frictionless. by the cable-pulley system shown in the figure. oth particles are constrained to vertical motion with particle able to slide Find: on a smooth You are vertical asked rod. here The to find system the speed is released of particle at s = when 0 m with has reached travelingthe downward with a speed of position v 1. ssume of s the pulleys to be small, massless and frictionless. = 2 m. Find: Find the speed of particle when has reached the position of s. 1.5 m s s g smooth rod Use the following parameters in your analysis: m = 10 kg, m = 10 kg, v 1 = 5 m/s and s = 2 m. IV-9

10 Chapter IV: Particle Kinetics Homework Homework 4.H Given: locks and (having masses of 2m and m, respectively) are connected by rigid, massless rod of length L. lock is constrained to move along a smooth vertical guide, and moves along a smooth, horizontal surface. The system is released from rest when is at a height of h 1 VE the path of. Homework Problems ME Find: Determine the speed of block when has dropped to a position that is at a distance of cks and (having masses of 2m and m, respectively) are connected by h 2 ELW the path of. id, massless rod of length L. lock is constrained to move along a ooth vertical guide, and moves ng a smooth, horizontal surface. e system is released from rest when 2m s at a height of h 1 VE the path. r this problem, ) Draw a free body diagram of the system made up of block, block and rod. g m L h 1 ) Determine the speed of block when has dropped to a position that is at a distance of h 2 ELW the path of. ysis, use the following parameter 10 kg, k = 1000 N/m, L = meters and h 2 = 0.3 meters. Use the following parameters in your analysis: m = 10 kg, L = 0.5 m, h 1 = 0.4 m and h 2 = 0.3 m. IV-10

11 Chapter IV: Particle Kinetics Homework inetics Homework Problems ME 274 Homework 4.I m IV-9 Given: The collar shown below, of mass m is pushed against a spring with stiffness k until it is llar, shown below, compressed of mass m and is pushed then released. against a The spring coefficient with stiffness of kinetic k until friction it is between the collar and rod is ssed!, and then µ k. released. ssuming that there is a kinetic coefficient of friction een the collar and rod, and that the collar fails to launch clear of the rod, ine the maximum Find: height Determine obtained the by maximum the collar, height as obtained measured byfrom the collar the point as measured of from the point of release.. ssume that the collar remains on the rod at all times. g m µ k k! following: m Use = 2.5 the kg, following k = 2000 parameters N/m,! = 0.3 in your m, µ analysis: k = 0.20, and m =! 4= kg, k = 2500 N/m,, = 0.01 m, µ k = 0.15 and θ = 30. IV-11

12 etics Homework Problems ME 274 Chapter IV: Particle Kinetics Homework IV-8 Homework 4.J r, shown below, Given: of mass The m, collar, starts shown from rest below, at point of mass. m, constant starts from force restf at is point. constant force F is the collar in applied the direction to theshown. collar indetermine the direction the shown. speed of Note the that collar the when mechanism it lies in the vertical plane. oint. Note that ssume the mechanism all surfaceslies to be in smooth. the vertical plane. ssume all surfaces oth. Find: Determine the speed of the collar when it reaches point. F m g r ollowing: mg = Use 6 lb, the F = following 30 lb and parameters r = 1.5 ft. in your analysis: mg = 10 lb, F = 6 lb, and r = 2 ft. IV-12

13 Problem IV-11 Pellets and (having masses m and 3m, respectively) are placed within a smooth tube trapping a small compartment of fuel. t a time when the pellets are initially at rest, the Chapter fuel is ignited. IV: Particle The Kinetics combustion Homework occurs over a short time!t, and over this time the combustion applies equal and opposite forces on the pellets with this force idealized by Homework the force time 4.Khistory F(t) shown below. a) Draw individual free body diagrams (FD s) for pellets and. From these Given: Pellets FD s, determine and (having the speed masses of meach andpellet 3m, at respectively) a time t are placed within a smooth tube 2, where t 2 >!t. trapping a small compartment of fuel. t a time when the pellets are initially at rest, the fuel is ignited. b) The Draw combustion an FD for occurs pellets over and a short together time t, (i.e., and treat over this and time as a the single combustion system). applies equal andfrom opposite this FD, forcesprovide on the an pellets argument with this that linear force idealized momentum by for the the force system timeof history F (t) shown below. and together should be conserved. Use your results from a) above to verify this argument. Find: Determine the speed of each pellet at time t 2, where t 2 > t. F(t) F 0 smooth m 3m smooth combusting fuel 0!t t Use the following parameters in your analysis: F Use the following parameters in your analysis: 0 = 3000 N, t = s and m = 0.75 kg. F 0 = 2000 N,! t = s and m = 0.5 kg. IV-13

14 Problem IV-12 particle of mass m is dropped from rest when at a height h 1 above a rigid floor. The particle impacts the floor with a speed of v 1. This impact of the particle with the floor Chapter IV: Particle Kinetics Homework lasts for a short duration of time!t, and after the impact is complete, the particle rebounds upward with a speed of v 2. The particle continues upward reaching a maximum Homework height 4.L of h 2. Given: a) particle Draw of a mass free body m isdiagram dropped(fd) from rest of the when particle at a for height the htime 1 above period a rigid during floor. its The particle impacts impact the floor with with the floor. a speed of v 1. This impact of the particle with the floor lasts for a short duration b) ased of time on t, your andfd after in the a), determine impact is complete, the average theforce particle acting rebounds on the particle upward by with a speed of v 2. The the particle floor during continues impact. upward Recall reaching that the aaverage maximum value height of a of force h 2. F(t) acting over a Find: For thisperiod problem: of time!t is given by: F ave = 1! t (a) Determine the average force acting on the particle! t! F( t) dt. by the floor during impact in the presence 0 of gravity; (b) Determine c) Repeat the average your calculations force acting in onb) the above particle but now by the ignore floorthe during weight impact force induring the absence the of gravity; time of impact. Did your answer for the average impact force change much? (c) Compare d) your Determine answers the from value (a) of and h (b); 2 / h 1. (d) Determine the value of h 2 /h 1. v = 0 v = 0 h 1 h 2 v = v 2 v = v 1 Use the following parameters in your analysis: t = s, m = 15 kg, v 1 = 80 m/s and v 2 = 50 m/s. Use the following parameters in your analysis:! t = s, m = 10 kg, v 1 = 60 meters / sec and v 2 = 45 meters / sec. IV-14

15 Chapter IV: Particle Kinetics Homework Particle Kinetics Homework Problems ME 274 Homework Problem 4.M IV-14 cannonball P of mass m is fired toward a steel barrier on a stationary cart. t some Given: cannonball P of mass m is fired toward a steel barrier on a stationary cart. t some time after rebounding time after from rebounding the barrier, from the the cannonball barrier, the is cannonball observed to is have observed a speed to of have v P and a speed is moving of v P in the direction and moving shownin below a direction in the figure. shown Let below M in bethe thefigure. combined Let mass M be of the the combined cannon/cart. mass ssume of the that the cannon cart isand ablecart. to move ssume without that the friction cart is along able the to move horizontal without surface friction andalong ignorethe the influence of air resistance. horizontal surface and ignore the influence of air resistance. Find: For a) thisdetermine problem: the velocity (both magnitude ND direction) of the cart after the (a) Determinecannonball the velocitybounces vector off the the cart steel after barrier theat cannonball the instant bounces shown below. off the steel barrier at the instant b) Let shown!t represents below; the elapsed time between the firing of the cannonball and the (a) If t represents instant the shown elapsed below. timedetermine between the firing average of value the cannonball of the horizontal and theforce instant acting shown below, determine on the combined the averagecannon value of and the cart horizontal over the force time acting period on of the 0 < combined t <!t. cannon/cart over the time period of 0 < t < t. P v P! m cannon cart M Use the following parameters in your analysis: mg = 60 lb, Mg = 200 lb, t = 0.2 s, θ = 20 and v P = 120 ft/s. Use the following parameters in your analysis: mg = 50 lb, Mg = 250 lb,! t = 0.4 s,! = 25 and v P = 100 ft / s. IV-15

16 Problem IV-50 Given: smooth, circular slot is cut into block with block 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 Chapter at IV: the Particle right edge Kinetics of the Homework block, as shown in the figures below. In Position 1, block is stationary, and a particle is released from rest into the upper opening of the slot. t Position 2 shown Homework 4.N below, particle is exiting the slot at the right edge of the block. The masses of and are M = 40 kg and m = 20 kg, respectively. The radius of the Given: smooth, circular circular slot slot is is cut given intoby block r = 2 meters. with block 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 is stationary, Find: and a particle Determine is released the velocity from rest of block into the and upper the velocity opening of of particle the slot. t at position Position 2. 2 shown below, particle is exiting the slot at the right edge of the block. The masses of and are M andyou m, respectively. must provide The an accurate radius offree thebody circular diagram slot isof r. the system used in your solution in order to receive full credit for your work. Consider using both the linear impulsemomentum the velocity and work-energy of block equations and the velocity in solving of this particle problem. at position Find: Determine 2. m V 2 M r g V 2 PSITIN 1 PSITIN 2 Use the following parameters in your analysis: m = 20 kg, M = 40 kg and r = 2 m. IV-16

17 Chapter IV: Particle Kinetics Homework Homework 4. Particle Kinetics Homework Problems ME 274 Given: particle of mass m is projected horizontally to the right at a height h above a smooth, horizontal floorproblem with a speed IV-15 of v 0. The particle strikes the floor at a horizontal distance d from where it was initially particle projected. of mass m The is projected coefficient horizontally of restitution to the right of at the a height impact h above of a the smooth, particle with the horizontal floor with a speed of v floor is e. 0. The particle strikes the floor at a horizontal distance d from where it was initially projected. The coefficient of restitution of the impact of the particle with the floor is e. Find: For this problem: a) Determine the angle! (a) Determine the angle θ 1 that the velocity 1 that the velocity of the particle has with the horizontal of the particle has with the horizontal immediately before impact. Use conservation of energy prior to impact. before impact; (b) Determine the b) angle Determine θ 2 that the angle the velocity! 2 that the of velocity the particle of the particle has has with with the the horizontal immediately immediately after impact. Use conservation of linear momentum in the horizontal after impact. direction along with the coefficient of restitution equation during impact. v 0 h v 2! 2 d EFRE impact v 1! 1 immediately FTER impact Use the following parameters in your analysis: e = 0.6, h = 125 ft and v 0 = 150 / sec. Use the following parameters in your analysis: e = 0.5, h = 150 ft and v 0 = 160 ft/s. IV-17

18 s and (having masses of m and M, respectively) are initially Chapter IV: Particle Kinetics Homework ing in directions perpendicular to each other with speeds of v 1 and v 1, tively, as shown Homework below in the 4.Pfigure. fter impacting each other, is ing to the RIGHT with a speed of v 2, and travels with a speed of v 2. der all surfaces Given: to be smooth. locks and (having masses of m and M, respectively) are initially traveling in directions perpendicular to each other with speeds of v 1 and v 1, respectively, as shown below in the figure. fter impacting each other, is traveling to the RIGHT with a speed of v 2, and travels is problem, with a speed of v 2 (the direction of motion for after impact is not known). Consider all surfaces ) draw FD s tofor be smooth. alone, alone and +. Include a set of coordinate axes with these FD s. Find: For this problem: ) determine the mass M of block. (a) Determine the mass M of block ; ) determine the (b) coefficient Determineof the restitution, coefficient e, of for restitution the impact eof for the and impact. of and. write down the linear momentum vector ns for, and + in terms of their and tangential components, as well as ficient of restitution equation. Use these ns to find M and e. v 1 m M g parameters in your analysis: m = 3kg, eters / sec, v 2 = 2 meters / sec rs / sec. v 1 HRIZNTL PLNE Use the following parameters in your analysis: m = 3 kg, v 1 = 4 m/s, v 1 = 4 m/s, v 2 = 2 m/s and v 2 = 5 m/s. IV-18

19 Problem IV-17 satellite P of mass m is in orbit around a planet whose center is at E. t position 1 shown below, the satellite is at a distance R Chapter IV: Particle Kinetics Homework 1 from E and moving with a speed of v 1. t position 2 shown, P is at a distance of R 2 from E and moving with a speed of v 2 with the velocity of P oriented as shown below. ssume that the only force acting on the satellite is a Homework gravitational 4.Q force directed toward E, and that the planet is not accelerating. Given: a) Determine satellite the angular P of mass momentum m is in orbit of P around about E aat planet positions whose 1 and center 2. is at E. t position 1 shown b) Provide below, an theargument satellite is to at support a distance the claim R 1 from that Ethe and angular moving momentum with a speed about ofe v 1 is. t position 2 shown, constant. P is at a distance of R 2 from E and moving with a speed of v 2 with the velocity of P oriented as shown below. ssume that the only force acting on the satellite is a gravitational force directed c) ased toward on b) E, above, and that determine the planet the speed is stationary. of P position 2, v 2, in terms of R 1, R 2,! 2 and v 1. Find: For this problem: d) Explain why your result in c) does not depend on the gravitational force acting on (a) Determine the speed of P at position 2, v 2, in terms of R 1, R 2, θ 2 and v 1 ; the satellite. (b) Provide an argument explaining why your result in (a) does not depend on the gravitational force acting on the satellite. e! e R v 2! 2 e! R 2 P v 1 E R 1 P e R IV-19

20 Chapter IV: Particle Kinetics Homework Homework 4.R ics Homework Problems ME 274 Given: Rigid arm (having length L and having negligible mass) is pinned to ground at end. particle of mass M is attached to end of. t instant 1, a pellet P (having a mass of m) strikes the stationary particle with a speed of v P 1 in the direction shown below in the figure. t IV-19 the end of a short time interval impact, P sticks to. (having length L and having negligible mass) is pinned to ground at end. of mass M is attached Find: Determine to end of the. angular t instant speed 1 of arm a pellet P immediately (having a mass after P sticks to. es the stationary particle with a speed of v P1 in the direction shown below in t the end of a short time interval ticks to. w a free body diagram (FD) of the tem made up of particles and P during act. Recall that P and are particles., the physical dimensions of these two ies are negligible). ased on your FD, vide argument that supports the claim t the angular momentum for system P bout is conserved during impact. termine the angular speed of arm immediately after P sticks to. nsider using the angular momentum ation. g! L m P M v P1 VERTICL PLNE Use the following parameters in your analysis: φ = 30, L = 4 ft, mg = 4 lb, Mg = 8 lb and v P 1 llowing parameters = 150in ft/s. your analysis:! = 36.87, L = 5 ft, mg = 2 lb, and v P1 = 120 ft / s. IV-20

21 Chapter IV: Particle Kinetics Homework Homework 4.S Particle Kinetics Homework Problems ME 274 Problem IV-41 Given: Particles and (having masses of 2m and m, respectively) are attached to a rod with Particles and (having massess of 2m and m, respectively) are attached to a rod with a a length of length L and of L with and with negligible negligible mass. mass. The The bar bar is is released released from from rest at rest when at it is when at an it is at an angle of θ 1 from the angle vertical, of asfrom shown the vertical, below as forshown Position below 1. for t Position the instant 1. t the instant when when the bar the passes through the vertical bar position passes through (Position the vertical 2), particle position, particle is released is released from from the the bar, with particle remaining attached to the remaining bar. attached to the bar. Find the maximum angle! reached by the bar after particle is released. Clearly state Find: Findthe the systems maximum used in your angle calculations θ 3 reached during byall the phases bar of after the motion. particle Use m is= released. 10 kg and L = 3 meters. Position 1 Position 2 Position 3 L/2 L/2 m! 1 2m! 3 is released at this position Use the following parameters in your analysis: θ 1 = 56.13, m = 10 kg and L = 3 m. IV-21

22 Particle Kinetics Homework Problems ME 274 Problem IV-51 Chapter IV: Particle Kinetics Homework Given: particle P, having a mass of m = 10 kg, is free to slide on a smooth, Homework 4.T lightweight bar. The bar is free to rotate in a horizontal plane about a vertical shaft passing through. spring, having a stiffness k = 400 N/m and Given: particleunstretched P, having alength mass of R 0 m, = 2 ismeters, free to is slide connected a smooth, between lightweight P and. The bar. spring The bar is free to rotate in is a horizontal compressed plane to half about of its a vertical unstretched shaftlength passing and through released. when spring, the bar having has a stiffness k and unstretched a rotational length speed R 0, of is! connected 1 = 5 rad/sec, between as shown P andin. the The figure spring below is compressed left. to half of its unstretched length and released when the bar has a rotational speed of ω 1, as shown in the figure below left. Find: For the position when R = R 0 (shown in the figure below right): Find: For the position a) when Determine R = Rthe 0 (shown rotation inrate the figure! 2 of the below bar. right): (a) Determine the rotation rate ω 2 of the bar; b) Determine the value of dr (b) Determine the value of Ṙ. dt. P R = R 0 /2! 1 k! 2 k P R = R 0 HRIZNTL PLNE Use the following parameters in your analysis: m = 10 kg, k = 400 N/m, R 0 = 2 m and ω 1 = 5 rad/s. IV-22