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chap_I_hwk - Chapter I: Particle Kinematics Homework...

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Unformatted text preview: Chapter I: Particle Kinematics Homework Chapter I Particle Kinematics Homework I-1 I-2 Chapter I: Particle Kinematics Homework Chapter I: Particle article Kinematics Homework Problems Kinematics Homework I-3 ME 274 roblem I/1 Problem I.1 √ 2 article P moves within the x-y plane on awithin the x-y plane (on = 2 4 xdescribed by y (x) = 2 4x2 + 12 where Given: Particle P moves path described by y x ) a path + 12 , where oth x and y areboth x in feet.are given in feet. given and y Find: Determine the acceleration of as a vector in terms as a x-y etermine the acceleration of P. Write your answer P. Write your answerof itsvector in terms of its x-y components. omponents. y path of P P x U parameters in parameters in your analysis: t sec x = ˙ ¨ ! se the followingse the followingyour analysis: x = 6 ft , x = !2x f=/2f t, and −10f t/sec and x = 0 ! = !3 ft / sec 2 . I-4 Chapter I: Particle Kinematics Homework Problem I.2 Particle Kinematics Homework Problems ME 274 Given: An automobile A is traveling on a circular path centered at O and having a radius of R. The automobile has a speed of v and is changing this speed at a rate of v . ˙ Problem I/2 Find: For this on a circular path centered at O and having a radius of R. An automobile A is traveling problem: ! The automobile has a speed of v , and is changing this speed at a rate of v . • Determine the acceleration of A. Write this as a vector in terms of its x-y components. a) Determine the acceleration of A. Write this as a vector in terms of its x-y • Make a sketch of the acceleration vector for A. components. • Determine the magnitude of for A. b) Make a sketch of the acceleration vector the acceleration of A in terms of the number of ”g’s” experienced by a passenger in the automobile. c) Determine the magnitude of the acceleration of A in terms of the number of “g’s” experienced by a passenger in the automobile. y v A R ! x O Use the following parameters in your analysis: 53.13 75 v 20 m / s ◦ nd ˙ Use the following parameters in your analysis: R = 150 m , ! = R = ° , m,= θ = 135a, v = 10m/sec and v = ! v = !2 m / s 2 . −6m/sec2 . Chapter I: Particle Kinematics Homework I-5 Problem I.3 Given: A particle P travels in the x-y plane with a path whose coordinates are given as a function of time t as: x (t) = 16 − 12t and y (t) = 2 + 15t − 3t2 . Find: For this problem: • Determine the velocity and acceleration of P in terms of their x-y components. • Make a sketch of the velocity and acceleration vectors for P. • Determine the rate of change of speed of P and the radius of curvature for the path of P. Use the following parameters in your analysis: t = 10seconds. I-6 Chapter I: Particle Kinematics Homework Problem I.4 Given: A particle P moves within a plane with a path given in terms of the polar coordinates of: R = 2θ2 where R and θ are as defined below in the figure, and with R and θ given in meters and in radians, respectively. mework Problems ME 274 Find: For this problem: • Determine the velocity and acceleration vectors of P in terms of their polar coordinates. s within a plane with a path given in terms of the polar coordinates of: • Make a sketch of the velocity and acceleration vectors for P. • Determine the rate of change of speed of P. • Is speed of and with R nd ! given in meters and in e as defined below in the figure P increasingaor decreasing at this instant? Explain. ely. the velocity and acceleration vectors of P in eir polar coordinates. eR etch of the velocity and acceleration vectors P the rate of change of speed of P. f P increasing or decreasing at this instant? e! ! R O ˙ ¨ Use the following parameters in your analysis: θ = π /2radians, θ = −3rad/sec and θ = −2rad/sec2 . ! parameters in your analysis: ! = " radians , ! = " 1.5 rad / sec and Chapter I: Particle Kinematics Homework I-7 Problem I.5 Given: Automobile A is traveling down a roadway with a speed of vA when it encounters a hailstorm. Hailstone P is known to be falling with a speed of vP at an angle of θ forward of the automobile. Particle Kinematics Homework Problems ME 274 Find: For this problem: • Write down, as a vector, the velocity of P as seen by a passenger in the automobile. Problem I/5 Automobile A is traveling doangle roadway with a speed ofobserved velocity of P? a • At what wn a with the vertical is this vA when it encounters hailstorm. Hailstone P is known to be falling with a speed of vP at an angle of ! forward of the automobile. a) Write down, as a vector, the velocity of P as seen by a passenger in the automobile. b) At what angle with the vertical is this observed velocity of P? vP vA ! P A Use the following parameters in your analysis: vA = 40m/sec, vP = 10m/sec and θ = 20◦ . Use the following parameters in your analysis: vA = 20 m / sec , vP = 6 m / sec and ! = 53.13° . be small. termine the speed of block B when s A = 0 . I-8 termine the speed of block B when s A = 4 m . Chapter I: Particle Kinematics Homework Problem I.6 Given: Blocks A and B are connected by the pulley-system shown below. Block A moves downward with a constant speed of vA on a vertical guide. Assume the radii of the pulleys to be small. Find: For this problem: • Determine the speed of block B when sA = 0. • Determine the speed of block B when sA = 4m. 8 m / sec in your analysis. 1.5 m O sA sB A vA Use the following parameters in your analysis: vA = 25m/sec. B Chapter I: Particle Kinematics Homework I-9 Problem I.7 Given: Pin P moves within slots cut into links A and B. The horizontal position of link A is given by x = 20 + t2 /4 whereas the vertical position of link B is given by y = 15 − t3 /6, where t is given in seconds and with x and y given in mm. omework Problems ME 274 Find: Determine the velocity and acceleration of P at t = 2 seconds. roblem 2/66) thin slots cut into links A and B. The ion of link A is given by x = 20 + t 2 / 4 tical position of link B is given by where t is given in seconds and with x m. Determine the velocity and and P at t = 2 seconds. I-10 Chapter I: Particle Kinematics Homework Problem I.8 Given: A pitcher throws a baseball with an inital speed of v0 at an angle of θ with the horizontal. Find: Determine the rate of change of speed and radius of curvature of the path when: • the ball is released by the pitcher. ework Problems blem 2/126) ME 274 • the ball reaches the maximum height of its path. baseball with an inital speed of ! with the horizontal. Determine the peed and radius of curvature of the v0 ! released by the pitcher. ches the maximum height of its path. ec and ! = 30° in your analysis. Use the following parameters in your analysis: v0 = 100f t/sec and θ = 30◦ . Chapter I: Particle Kinematics Homework I-11 Problem I.9 Given: A string is used to pull in particle P in such a way that the radial position of P is given by r = 0.8 − 0.1t − 0.05t2 while the angular orientation of arm OA is given by θ = 0.4 + 0.12t + 0.06t2 , where r, θ and t are given in meters, radians and seconds, respectively. work Problems ME 274 Find: Determine the velocity and acceleration of P. lem 2/155) ull in particle P in such a way that the is given by r = 0.8 ! 0.1t ! 0.05t 2 while ion of arm OA is given by 06t 3 , where r, ! and t are given in seconds, respectively. Determine the ation of P when t = 2 seconds. Use the following parameters in your analysis: t = 2seconds. I-12 Chapter I: Particle Kinematics Homework Problem I.10 mework Problems Given: Block A moves to the right with a speed of vA . ME 274 Find: Determine the speed of block B. roblem 2/213) o the right with a speed of 3.6 ft/sec. eed of block B. Use the following parameters in your analysis: vA = 3.6f t/sec. Chapter I: Particle Kinematics Homework I-13 Problem I.11 mework Problems Given: Block B moves downward with a constant speed of vB . ME 274 Find: Determine the speed of block A as a function of the position y of block A. roblem 2/220) nward with a speed of vB . Determine k A as a function of the position y of I-14 Chapter I: Particle Kinematics Homework Problem I.12 Given: A skier is moving down a slope. Along the straight slope from A to B, her speed is changing Particle Kinematics Homework Problems at a constant rate of vAB . From B to C, the slope is curved with a radius of curvature of ρ. At ˙ point C, her speed is changing at a rate of vC . The skiers speed at points A, B and C are vA , vB ˙ Problem I/12 and vC , respectively. ME 274 A skier is moving down a slope. Along the straight slope from A to B, her speed is increasing at a constant rate of 2.4 m/s2. From B to C, the slope is curved with a radius of Find: For this problem: curvature of ! =150 m. At point C, her speed is decreasing at a rate of 2 m/s2. The skier’s peed at points A B and vectors m/s, 25 A, and 30 m/s, respectively. • Sketch the sunit tangent and, normal C are 15at points m/s B and C. a) Sketch the unit tangent and normal vectors at points A, B and C. • Calculate the accelerationthe acceleration vectors and C. A, B and C. b) Sketch vectors at points A, B at points c) Calculate the acceleration vectors at • Sketch the acceleration vectors at points A, B and C. points A, B and C. A B ! C Use the following parameters in your analysis: vAB = 5m/sec2 , vC = −1.5/sec2 , ρ = 175m, ˙ ˙ vA = 20m/sec, vB = 30m/sec and vC = 40m/sec Particle Kinematics Homework Problems ME 27 Problem I/13 I-15 M PartParticle Kinematics Homework Problems I A roller coaster car travels along a short part of the track given by the equation Problem I.13 Problem m a r = 50 cos(2! ) I/13nd ! = 0.02t 2 radians. What is the velocity and acceleration of the car P! = when art I20°? PART I A roller coaster car travelscoaster a short part of theshort part of the track given byrthe equation A roller along car travels along a track given by the equation = 50cos (2θ), where r is in meters and θ is in radians. What 2 the velocity and acceleration of the r = 50 cos(2! ) m and ! = 0.02t is radians. What is the velocity and acceleration of the car when θ = 10◦ ? when ! = 20°? Chapter I: Particle Kinematics Homework ! r ! r Part II Along a circular loop at the bottom of the track, the cars will move at a speed of vo with a ! rate of v . To ensure the safety of the passengers, the magnitude of the acceleration PART II Along a circular loop at the bottom of the track, the cars will move at a speed of v0 that Part II s.hould not exceed 4g. What is the minimum magnituderadius, acceleration of the track? is changing at a rate of v To ensure the safety of the passengers, the allowable of the R, of this part ˙ Along a circular loop at the bottom of the track, the cars will move at a speed of vo w should not exceed 4g. What is the minimum allowable radius, R, of this part of the track? ! Use ratefollowing ensure the safety of analysis: vo = 20 m/s,agnitudem/sthe acceleration the of v . To parameters in your the passengers, the m v = 15 of 2. ! should not exceed 4g. What is the minimum allowable radius, R, of this part of the tr ! R Use the following parameters in your analysis: vo = 20 m/s, v = 15 m/s2. R Use the following parameters in your analysis: v0 = 40m/sec and v = 10m/sec2 ˙ I-16 Chapter I: Particle Kinematics Homework Particle Kinematics Homework Problems ME 2 Problem I.14 Problem I/14 Two cars are travelling as shown. Car A is traveling along a circular path of radius, r, Given: Two cars are travelling as shown. Car A is traveling along a circular path of radius, r, with a constant speed, v A . Car B is traveling with a speed, vB , in the positive x axis. Th with a constant speed, vA . Car B is traveling with a speed, vB , in the positive x axis. The speed ! of B is changing at a ratespeed of B is changing at a rate of vB . At this instant, find: of vB . ˙ a) The velocity of A as seen by car B. Find: At this instant: b) The acceleration of A as seen by car B. • Determine the velocity of A as seen by car B. Use the following parameters in your analysis: r = 100 m, v A = 15 m/s, vB = 20 m/s, • Determine the acceleration m/s2. as seen by car B. ! vB = -2 of A B y A r x Use the following parameters in your analysis: r = 150m, vA = 20m/sec, vB = 30m/sec and vB = 4m/sec2 ˙ Chapter I: Particle Kinematics Homework I-17 Particle Kinematics Homework Problems Problem I.15 ME 274 Problem I/15 A worker is lifting a large crate, P, using a pulley shown in the diagram. He is holding Given: A worker tis lifting at point crate, P, using a pulley shownat athe diagram. He v .holdingis the speed he rope a large A and is walking to the right in constant speed is What the A rope at point A and is walking to the right at a constant speed vA . of the crate when the worker has walked a distance x? Find: What is the speed of the crate when the worker has walked a distance x? Use the following parameters in your analysis: v A = 1.5 m/s, x = 2 m, h = 5 m. O h P’ P A A’ x Use the following parameters in your analysis: vA = 2m/sec, x = 3m and h = 8m I-18 Chapter I: Particle Kinematics Homework Particle Kinematics Homework Problems ME 274 Problem I/16 Problem I.16 Particle P travels on a circular path having a radius of 2.5 feet with its center at point A, Given: Particle P travels on a circularseen by an observer at point with its center at point A, as as shown in Figure (a). This particle is path having a radius of L O that is 2 feet shown in Figure (a). This particle is seen by an observer at point O that is a distance of d directly directly to the left of point A. At the position shown in Figure (b) (! = 0), P is directly to the left of point A. At the position shown in Figure (b) (θ = 0), P is directly below point A, has below point A, has a speed of 10 ft/sec and has an acceleration that points directly toward a speed of vP and has an acceleration that points directly toward point A. point A. Find: In Figure (b) (b) shown below: i) For position below, make a sketch of the path unit vectors for P ( et and en ) and the polar unit of the path unit ! . • Make a sketch vectors e R and evectors for P (￿t and ￿n ) and the polar unit vectors ￿R and e e e ￿φ . e ii) For the position in Figure (b), find numerical values for the rate of change of speed • Determinefor R, !, R values .for the rate of change of speed of P. !!! ! of P and numerical! and !! ¨ ˙¨˙ • Determine numerical values for R, R, φ and φ. d O O A " R A R L aP ! P P (a) vP (b) Use the following parameters in your analysis: L = 2.5f t, d = 2f t and vP = 10f t/sec Particle Kinematics Homework Problems ME 274 Chapter I: Particle Kinematics Homework I-19 Problem I/17 Given: An inextensible cable is attached to block B, is wrapped around a pulleys C Problem I.17 and D and is attached to a fixed support at E. Block B is constrained to move on a vertical guide, and block A is constrained to slide along a horizontal Given: An inextensible cable is attached to block B, is wrapped around a pulleys C and D and surface. When sB = 4 ft , block B is moving downward with a speed of 15 is attached to a fixed support at E. Block B is constrained to move on a vertical guide, and block ft/sec A is constrained .to slide along a horizontal surface. Block B is moving downward with a constant speed of vB . Find: Determine the speed of block A at this instant. Find: Determine the speed of block A. sA D d C A sB E B vB Use the following parameters in your analysis: d = 3f t, sB = 4f t and vB = 15f t/sec I-20 Chapter I: Particle Kinematics Homework Problem I.18 Given: Particle P travels within the x-y plane along a path given by y (x) = x2 /2 − 10x, where x and y are given in feet. The y-component of the position for P is changing at a constant rate of y . ˙ Find: For this problem: • determine the velocity vector of P. • determine the acceleration vector of P. • determine the rate of change of speed of P. Use the following parameters in your analysis: y = 10f t/sec and x = 9f t. ˙ Chapter I: Particle Kinematics Homework I-21 Problem I.19 Given: A projectile is launched from ground with a speed of v0 at a angle of θ with respect to the horizontal. After firing, it can be assumed that only the gravitational force acts on the projectile; i.e., the acceleration of the particle is straight down. Particle Kinematics Homework Problems ME 274 Find: Considering the motion of the projectile at a time t: a) determine the radius of curvature Problem I/19 of its path, and b) determine the rate of change of speed of the projectile. g v0 ! Use the following parameters in your analysis: v0 = 450m/sec, θ = 53.13◦ and t = 25sec. I-22 Chapter I: Particle Kinematics Homework Problem I.20 Given: At the instant shown, the Cartesian components for the position of particle P are known as (xP , yP ). In addition, the Cartesian form for the velocity and acceleration of P are known as ￿P v and ￿ P , respectively. a Particle Kinematics Homework Problems ME 274 Find: For this instant: a) write down the polar unit vectors eR and eφ in terms of their Cartesian ˆ ˆ Problem I/20 ¨ ˙˙¨ components, and b) determine numerical values for R, R, φ, R and φ . ˆ eR P yP ! ˆ e! R xP ￿ ￿ Use the following parameters in your analysis: xP = 15m, yP = 20m, ￿P = −10ˆ + 20ˆ m/sec v i j ￿ ￿ and ￿ P = −8ˆ m/sec2 . a j Chapter I: Particle Kinematics Homework I-23 Problem I.21 Given: At the instant shown, the polar components for the velocity and acceleration of particle P are known as ￿P and ￿ P , respectively. v a Particle Kinematics Homework Problems ME 274 Find: For this instant, determine the speed, rate of change of speed and radius of curvature for the path of P. I/21 Problem ˆ eR ˆ e! P R ! Use the following parameters in your analysis: ￿P = (30ˆR − 40ˆθ ) f t/sec and ￿ P = (−4ˆθ ) f t/sec2 . v e e a e ...
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