KINEMATICS OF PARTICLES PROBLEMS ON RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES

Transcription

1 KINEMATICS OF PARTICLES PROBLEMS ON RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES

2 1. The car A has a forward speed of 18 km/h and is accelerating at 3 m/s2. Determine the elocity and acceleration of the car relatie to obserer B who rides in a nonrotating chair on the Ferris wheel. The angular rate Ω= 3 re/min of the Ferris wheel is constant. (2/188)

3 A = 18 km/h, aa = 3 m/s2., A/B =?, aa/b =?, Ω= 3 re/min (constant) 2π radians 1 minute Ω = ω = 3 reolutions = = rad / s (constant ) minute 1 reolution 60 seconds B = Ω R = (9) = (m / s ) B = B cos 45i B sin 45 j = 2i 2 j ( m / s ) Ω -t y, Y Bx x n 45 By X B +t A & R = 0 abt = Ω 2 & =α = 0 Ω B abn = a B = = = m / s 2 R 9 ab = ab cos 45i ab sin 45 j = 0.627i j (m / s 2 )

4 A = 18 km/h, aa = 3 m/s2., A/B =?, aa/b =?, Ω= 3 re/min (constant) A =18 km / h = 5 m / s A = 5i (m / s ) -t Ω y Y Bx x a A = 3i (m / s 2 ) n 45 By X B +t A = B + A/ B 5i = 2i 2 j + A/ B A A/ B = 3i + 2 j (m / s ) A/ B = 3.61 m / s A B A/B

5 A = 18 km/h, aa = 3 m/s2., A/B =?, aa/b =?, Ω= 3 re/min (constant) Ω -t y abx x 45 ab n X aby +t aa aa a A = 3i (m / s 2 ) ab aa/b a B = 0.627i j (m / s 2 ) a A = ab + a A/ B, 3i = 0.627i j + a A/ B a A/ B = 3.627i j (m / s 2 ), a A/ B = 3.68 m / s 2

6 2. Airplane A is flying horizontally with a constant speed of 200 km/h and is towing the glider B, which is gaining altitude. If the tow cable has a length r = 60 m and θ is increasing at the constant rate of 5 degrees per second, determine the magnitudes of the elocity and acceleration of the glider for the instant when θ = 15. (2/196) a

7 A = 200 km/h (cst), r = 60 m, θ& = 5 deg/s (cst), determine magnitudes of elocity and acceleration of glider for θ = 15. A = 200 km / h = m / s θ& = 5 o = rad / s s +θ eθ e +r r B/A θ θ&& = 0 r& = &r& = 0 B = A + B / A B -θ ab=abr A = cos15er sin 15eθ = 53.67er eθ B / A = B / Aeθ, B / A = rθ& = 60 (0.087) = 5.22 m / s, B / A = 5.22eθ B = Br er + Bθ eθ B = 53.67er eθ eθ = 53.67er eθ (m / s ) B = m / s ( km / h) A θ A θ Ar -r B A ab = a A + ab / A, aa = 0 ab = ab / A, ab = ab r er + abθ eθ ab r = &r/& rθ& 2 = 60 (0.087) 2 = m / s 2, abθ = 2r&θ& + rθ&& = 0 ab = 0.454er (m / s 2 ) ( from B to A) B/A

8 3. After starting from the position marked with the x, a football receier B runs the slant-in pattern shown, making a cut at P and thereafter running with a constant speed B = 7 m/s in the direction shown. The quarterback releases the ball with a horizontal elocity of 30 m/s at the instant the receier passes point P. Determine the angle α at which the quarterback must throw the ball, and the elocity of the ball relatie to the receier when the ball is caught. Neglect any ertical motion of the ball. (2/203)

9 B= 7 m/s (cst), A= 30 m/s, determine angle α and A/B.

10 4. A batter hits the baseball A with an initial elocity of 0 = 30 m/s directly toward fielder B at an angle of 30 to the horizontal; the initial position of the ball is 0.9 m aboe ground leel. Fielder B requires ¼ s to judge where the ball should be caught and begins moing to that position with constant speed. Because of great experience, fielder B chooses his running speed so that he arries at the catch position simultaneously with the baseball. The catch position is the field location at which the ball altitude is 2.1 m. Determine the elocity of the ball relatie to the fielder at the instant the catch is made. (2/206)

11 0 = 30 m/s. Fielder B requires ¼ s to judge where the ball should be caught, then moes with constant speed, he arries at catch position (y=2.1 m) simultaneously with the baseball. Determine elocity of the ball relatie to fielder at the instant of catch. 1 2 A Ball y = y0 + 0 y t gt 2.1 = sin 30t 4.905t 2 2 ax + bx + c = t 15t = 0 t1 = 0.08 s t 2 = 2.98 s 2 x = x0 + 0 x t B Player 65 + x = cos 30 (2.98), x = m x = x0 + 0 x t 65/ = 65/ + B (2.73) t = t 0.25 = = 2.73 s, B = 4.55 m / s

12 0 = 30 m/s. Fielder B requires ¼ s to judge where the ball should be caught, then moes with constant speed, he arries at catch position (y=2.1 m) simultaneously with the baseball. Determine elocity of the ball relatie to fielder at the instant of catch. A = B + A / B Ax = A0 x = 30 cos 30 = m / s Ay = A0 y gt = 30 sin (2.98) = m / s A = Ax i Ay j = 25.98i j m / s, B = 4.55i A / B = 25.98i j 4.55i = 21.43i j (m / s )

13 5. Particles A and B both hae a speed of 8 m/s along the directions indicated by arrows. A moes in a curilinear path defined by y 2 = x 3 and B moes along a linear path defined by y = -x. If the elocity of B is decreasing at a rate of 6 m/s each second and the elocity of A is increasing at a rate of 5 m/s each second, determine the elocity and acceleration of A with respect to B for the instant represented.

14 A=B=8 m/s, elocity of B decreases at a rate of 6 m/s2, elocity of A increases at a rate of 5 m/s 2, determine A/B and aa/b. A B

15 6. Car A is traelling along a circular path haing a radius of 60 m with a constant speed of 54 km/h. At the instant A passes from the position indicated, car B is 30 m away from the junction and is increasing its speed of 72 km/h with 1.5 m/s2. For the instant represented, determine the elocity and acceleration of A with respect to an obserer at B. Also determine r, r&, &r&, θ, θ&, θ&& for this instant.

16 A traelling along a circular path of R = 60 m, A = 54 km/h (cst). At the instant A passes from the position indicated, car B is 30 m away from the junction and is increasing its speed of 72 km/h with 1.5 m/s2. determine elocity and acceleration of A with respect to an obserer at B. Also determine r, r&, &r&, θ, θ&, θ&&.