RELATIVE-MOTION ANALYSIS OF TWO PARTICLES USING TRANSLATING AXES Today’s Objectives: Students will be able to: Understand translating frames of reference Use translating frames of reference to analyze relative motion Dynamics, Fourteenth Edition R.C Hibbeler In-Class Activities: • Check Homework, • Reading Quiz • Applications • Relative Position, Velocity and Acceleration • Vector & Graphical Methods • Concept Quiz • Group Problem Solving • Attention Quiz Copyright ©2016 by Pearson Education, Inc All rights reserved READING QUIZ The velocity of B relative to A is defined as A) vB – vA B) vA – vB C) vB + vA D) vA + vB Since two-dimensional vector addition forms a triangle, there can be at most _ unknowns (either magnitudes and/or directions of the vectors) A) one B) two C) three D) four Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved APPLICATIONS When you try to hit a moving object, the position, velocity, and acceleration of the object all have to be accounted for by your mind You are smarter than you thought! Here, the boy on the ground is at d = 10 ft when the girl in the window throws the ball to him If the boy on the ground is running at a constant speed of ft/s, how fast should the ball be thrown? Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved APPLICATIONS (continued) When fighter jets take off or land on an aircraft carrier, the velocity of the carrier becomes an issue If the aircraft carrier is underway with a forward velocity of 50 km/hr and plane A takes off at a horizontal air speed of 200 km/hr (measured by someone on the water), how we find the velocity of the plane relative to the carrier? How would you find the same thing for airplane B? How does the wind impact this sort of situation? Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved RELATIVE POSITION (Section 12.10) The absolute positions of two particles A and B with respect to the fixed x, y, z-reference frame are given by rA and rB The position of B relative to A is represented by rB/A = rB – rA Therefore, if rB = (10 i + j ) m and rA = (4 i + j ) m, then rB/A = rB – rA = (6 i – j ) m Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved RELATIVE VELOCITY To determine the relative velocity of B with respect to A, the time derivative of the relative position equation is taken vB/A = vB – vA or vB = vA + vB/A In these equations, vB and vA are called absolute velocities and vB/A is the relative velocity of B with respect to A Note that vB/A = - vA/B Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved RELATIVE ACCELERATION The time derivative of the relative velocity equation yields a similar vector relationship between the absolute and relative accelerations of particles A and B These derivatives yield: aB/A = aB – aA or aB = aA + aB/A Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved SOLVING PROBLEMS Since the relative motion equations are vector equations, problems involving them may be solved in one of two ways For instance, the velocity vectors in vB = vA + vB/A could be written as two dimensional (2-D) Cartesian vectors and the resulting 2-D scalar component equations solved for up to two unknowns Alternatively, vector problems can be solved “graphically” by use of trigonometry This approach usually makes use of the law of sines or the law of cosines Could a CAD system be used to solve these types of problems? Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved LAWS OF SINES AND COSINES Since vector addition or subtraction forms a triangle, sine and cosine laws can b a be applied to solve for relative or absolute velocities and accelerations As A a review, their formulations are provided B c below b a c Law of Sines: sin A sin B sin C C Law of Cosines: a b c bc cos A 2 b a c ac cos B 2 a c b ab cos C Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved EXAMPLE Given: Find: Two aircraft as shown vA = 650 km/h vB = 800 km/h vB/A Plan:vectors vA and vB in 1) Vector Method: Write Cartesian form, then determine vB – vA 2) Graphical Method: Draw vectors vA and vB from a common point Apply the laws of sines and cosines to determine vB/A Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved EXAMPLE (continued) Solution: 1) Vector Method vA = (650 i ) km/h vB = –800 cos 60 i – 800 sin 60 j = ( –400 i – 692.8 j) km/h vB/A = vB – vA = (–1050 i – 692.8 j) km/h vB /A km/h = tan-1( ) = 33.4 Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved EXAMPLE (continued) 2) Graphical Method: Note that the vector that measures the tip of B relative to A is vB/A 650 km/h vA 800 km/h 120 vB vB/A Law of Cosines: (vB/A)2 = (800) + (650) − (800) (650) cos 120 vB/A = 1258 km/h Law of Sines: vB/A sin(120 ) Dynamics, Fourteenth Edition R.C Hibbeler = vA sin or Copyright ©2016 by Pearson Education, Inc All rights reserved CONCEPT QUIZ Two particles, A and B, are moving in the directions shown What should be the angle so that vB/A is minimum? A) 0° B) 180° C) 90° D) 270° vB = ft/s B vA= ft/s A Determine the velocity of plane A with respect to plane B A) (400 i + 520 j ) km/hr B) (1220 i - 300 j ) km/hr C) (-181 i - 300 j ) km/hr 30 D) (-1220 i + 300 j ) km/hr Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved GROUP PROBLEM SOLVING Given: Car A moves in a straight line while Car B moves along a curve having a radius of curvature of 200 m vA = 40 m/s vB = 30 m/s aA = m/s2 aB = -3 m/s2 Find: vB/A aB/A Plan: Write the velocity and acceleration vectors for Cars A and B Determine vB/A and aB/A by using vector relationships Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved GROUP PROBLEM SOLVING (continued) y Solution: The velocity of B is: vB = -30 sin(30) i + 30 cos(30) j = (-15 i + 25.98 j) m/s The velocity of A is: vA = 40 j (m/s) x The relative velocity of B with respect to A is (vB/A): vB/A = vB – vA = (-15 i + 25.98 j) – (40 j) = (-15 i – 14.02 j) m/s or vB/A = (-15)2 + (-14.02)2 = 20.5 m/s 14.02 = tan-1( 15 ) = 43.1° Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved GROUP PROBLEM SOLVING (continued) g a curve, the acceleration of B is: B = (at)B + (an)B = [-(-3) sin(30) i + (-3) cos(30) j] 302 + [- ( ) cos(30) i - ( ) sin(30) j ] 200 B 302 200 y = ( -2.397 i – 4.848 j ) m/s2 The acceleration of A is: aA = (4 j ) m/s2 x The relative acceleration of B with respect to A is: aB/A = aB – aA = ( -2.397 i – 8.848 j ) m/s2 aB/A = (-2.397)2 + (-8.848)2 = 9.17 m/s2 = tan-1(8.848 / 2.397) = 74.8° Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved ATTENTION QUIZ Determine the relative velocity of particle B with respect to particle A y A) (48i + 30j) km/h B v =100 km/h B) (- 48i + 30j ) km/h C) (48i - 30j ) km/h 30 x A v =60 km/h D) (- 48i - 30j ) km/h B A If theta equals 90° and A and B start moving from the same point, what is the magnitude of rB/A at t = s? A) B) C) D) 20 ft 15 ft 18 ft 25 ft Dynamics, Fourteenth Edition R.C Hibbeler B A vB ft/s vA ft/s Copyright ©2016 by Pearson Education, Inc All rights reserved End of the Lecture Let Learning Continue Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved ... (either magnitudes and/or directions of the vectors) A) one B) two C) three D) four Dynamics, Fourteenth Edition R. C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved APPLICATIONS... ©2016 by Pearson Education, Inc All rights reserved APPLICATIONS (continued) When fighter jets take off or land on an aircraft carrier, the velocity of the carrier becomes an issue If the aircraft... 2 b a c ac cos B 2 a c b ab cos C Dynamics, Fourteenth Edition R. C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved EXAMPLE Given: Find: Two aircraft as shown