CEE 271: Applied Mechanics II, Dynamics Lecture 6: Ch.12, Sec.10

Transcription

1 1 / 18 CEE 271: Applied Mechanics II, Dynamics Lecture 6: Ch.12, Sec.10 Prof. Albert S. Kim Civil and Environmental Engineering, University of Hawaii at Manoa Date:

2 2 / 18 RELATIVE-MOTION ANALYSIS OF TWO PARTICLES USING TRANSLATING AXES Today s objectives: Students will be able to 1 Understand translating frames of reference. 2 Use translating frames of reference to analyze relative motion. In-class activities: Reading Quiz Applications Relative Position, Velocity and Acceleration Vector and Graphical Methods Concept Quiz Group Problem Solving Attention Quiz

3 3 / 18 READING QUIZ 1 The velocity of B relative to A is defined as (a) v B v A (b) v A v B (c) v B + v A (d) v A + v B ANS: (a) 2 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 ANS: (b)

4 4 / 18 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 4 ft/s, how fast should the ball be thrown?

5 5 / 18 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 do 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?

6 6 / 18 RELATIVE POSITION (Section 12.10) The absolute position of two particles A and B with respect to the fixed x, y, z reference frame are given by r A and r B. The position of B relative to A is represented by r B/A = r B r A Therefore, if r B = (10i + 2j) m and r A = (4i + 5j) m, then r B/A = (6i 3j) m.

7 7 / 18 RELATIVE VELOCITY To determine the relative velocity of B with respect to A, the time derivative of the relative position equation is taken. or v B/A = v B v A v B = v A + v B/A In these equations, v B and v A are called absolute velocities and v B/A is the relative velocity of B with respect to A. Note that v B/A = v A/B.

8 8 / 18 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: a B/A = a B a A or a B = a A + a B/A

9 9 / 18 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 v B = v A + v B/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?

10 10 / 18 LAWS OF SINES AND COSINES Since vector addition or subtraction forms a triangle, sine and cosine laws can be applied to solve for relative or absolute velocities and accelerations. As a review, their formulations are provided below. Law of Sines: Law of Cosines: a sin A = b sin B = c sin C a 2 = b 2 + c 2 2bc cos A b 2 = a 2 + c 2 2ac cos B c 2 = a 2 + b 2 2ab cos C

11 11 / 18 EXAMPLE Given: v A = 650 km/h v B = 800 km/h Find: v B/A Plan: 1 Vector Method: Write vectors v A and v B in Cartesian form, then determine v B v A 2 Graphical Method: Draw vectors v A and v B from a common point. Apply the laws of sines and cosines to determine v B/A.

12 12 / 18 EXAMPLE (continued) Solution a). Vector Method: v A = 650i v B = 800 cos 60i 800 sin 60j = 400i 692.8j Q: Where is an origin? v B/A = v B v A = 1050i 692.8j (1) v B/A = ( 1050) 2 + ( 692.8) 2 = 1258 (2) ( ) θ = tan 1 = 33.4 (3) 1050

13 13 / 18 EXAMPLE (continued) Solution b). Graphical Method: Note that the vector that measures the tip of B relative to A is v B/A. Law of Cosines: (v B/A ) 2 = (800) 2 + (650) 2 (800)(650) cos 120 Law of Sines: v B/A = 1258 km/h v B/A sin 120 = v B sin θ θ = 33.4

14 CONCEPT QUIZ 1. Two particles, A and B, are moving in the directions shown. What should be the angle θ so that v B/A is minimum? (a) 0 (b) 180 (c) 90 (d) 270 ANS: (a) 2. Determine the velocity of plane A with respect to plane B. (a) (400i + 520j) km/hr (b) (1220i 300j) km/hr (c) ( 181i 300j) km/hr (d) ( 1220i + 300j) km/hr ANS: (b) 14 / 18

15 15 / 18 GROUP PROBLEM SOLVING Given: v A = 30 mi/h v B = 20 mi/h a A = 0 mi/h 2 a B = 1200 mi/h 2 Find: v B/A and a B/A Plan: Write the velocity and acceleration vectors for A and B and determine v B/A and a B/A by using vector equations.

16 16 / 18 GROUP PROBLEM SOLVING (solution continued) Solution: The velocity of B is: v B = 20 sin(30)i + 20 cos(30)j = ( 10i j) mi/h The velocity of A is: v A = 30i (mi/h) The relative velocity of B with respect to A is (v B/A ): v B/A = ( 10i j) ( 30i) = (20i j) mi/h v B/A = (20) 2 + (17.32) 2 = 26.5 mi/h θ = tan 1 ( ) = 40.9

17 17 / 18 GROUP PROBLEM SOLVING (solution continued) The acceleration of B is: a B = (a t ) B + (a n ) B = [ 1200 sin 30i cos 30j] + [ cos 30i sin 30j] = 554.7i j( mi/h 2 ) The acceleration of A is zero : a A = 0 The relative acceleration of B with respect to A is: a B/A = a B a A = 554.7i j( mi/h 2 ) (4) a B/A = (554.7) 2 + (1706) 2 = 1790 mi/h 2 (5) β = tan 1 ( ) = 72 (6)

18 18 / 18 ATTENTION QUIZ 1 Determine the relative velocity of particle B with respect to particle A. (a) (+48i + 30j) km/h (b) ( 48i + 30j) km/h (c) (+48i 30j) km/h (d) ( 48i 30j) km/h ANS: (c) 2 If θ equals 90 and A and B start moving from the same point, what is the magnitude of r B/A at t = 5 s? (a) 20 ft (b) 15 ft (c) 18 ft (d) 25 ft ANS: (d)