An aircraft P takes off at A with a velocity of 250 km/h and climbs in the vertical y'-z' plane at the constant 15° angle with an acceleration along its flight path of 0.8 m/s , Flight progress is monitored by radar at point O. to) Resolve the velocity of P into cylindrical-coord in ate components 60 seconds after takeoff and find r, 9, and z for that instant. (SO Resolve the velocity of the aircraft P into spherical-coordinate components 60 seconds after takeoff and find R, 9, and ij> for that instant.
Solution, (a) The accompanying figure shows the velocity and accel- . eration vectors in the y'-z' plane. The takeoff speed is
^u = Y j r = 69.4 m/s
and the speed after BO seconds is
The distance s traveled after takeoff is
s s„ + v0t + |at2 = 0 + 69.4(60) + \ (O.StteO)2 = 5610 m
The y-coordinate and associated angle if are y = 5610 cos 15° = 5420 m
f) = tan"1 = 61.0°
3000
From the figure (f>) of .t'-y projections, we have r = v>'3000s • 5420s = 6190 m
(b) Refer to the accompanying figure (c), which shows the .i-y plane and various velocity components projected into the vertical plane con-taining r and R. Note that
86 Chapter 2 K i n e m a t i c s of P a r t i c l e s
PROBLEMS
Introductory Problems
2/171 The rectangular coordinates of a particle are given in millimeters as functions of time t in seconds by x — 30 cos 2t,y = 40 sin 2f, and z = 20i + 3 r .
Deter-mine the angle between the position vector r and the velocity v and the angle th between the position vector r and the acceleration a, both at time t • 2 s.
Ans. (ti = 00.8°, t)2 = 122.4°
2/172 A projectile is launched from point O with an ini-tial velocity of magnitude v0 = 600 ft/sec, directed as shown in the figure. Compute the x-, y-, and
¿-components of position, velocity, and acceleration 20 seconds after launch. Neglect aerodynamic drag.
Problem 2/172
2/173 The particle P moves along the space curve and has a velocity v = 4i — 2j — k m/s for the instant shown. At the same instant the particle has an ac-celeration a whose magnitude is 8 m/s2. Calculate the radius of curvature p of the path for this posi-tion and the rate u at which the magnitude of the velocity is increasing.
Ans. p
= 7.67 m,v
= 7.52 m/s22/174
Problem 2/173
An amusement ride called the "corkscrew" takes the passengers through the upside-down curve of a horizontal cylindrical helix. The velocity of the car's as they pass position A is 15 m/s, and the compo-nent of their acceleration measured along the tan-gent to the path is g cos y at this point. The effective radius of the cylindrical helix is 5 m, and the helix angle is y = 40°. Compute the magnitude of the ac-celeration of the passengers as they pass position A.
A , yiotv'fc- —
5 m
Vert, '' y = 40°
Problem 2/152
Article 2/7 P r o b l e m s 8 7
Problem 2/176 Problem 2/17S
2/175 Ail industrial robot is being used to position a small part P. Calculate the magnitude of the acceleration a of P for the instant when p = 30° if p = 10 de-grees per second and fi 20 dede-grees per second squared at this same instant . The base of the robot is revolving at the constant rate uj = 40 degrees per second. During the motion arms AO and AP re-main perpendicular.
Ans. a = 219 mm/s
Representative Problems
2 / 1 7 7 The car A is ascending a parking-garage ramp in the form of a cylindrical helix of 24-ft radius rising 10 ft for each half turn. At the position shown the car has a speed of 15 mi/hr, which is decreasing at the rate of 2 mi/hr per second. Determine the r-, 11-, and ¿-components of the acceleration of the car.
A/is. ar = - 1 9 . 8 2 ft/sec2
aj = - 2 . 9 1 ft/sec2
a, = - 0 . 3 8 6 ft/sec2
2 / 1 7 6 The rotating element in a mixing chamber is given a periodic axial movement z = ¿u sin 2nmt while it is rotating at the constant angular velocity Q = to. De-termine the expression for the maximum magnitude of the acceleration of a point A on the rim of radius r. The frequency n of vertical oscillation is constant.
z
Problem 2/177
2 / 1 7 8 An aircraft takes off at A and climbs at a steady angle with slope of 1 to 2 in the vertical y-z plane at a constant speed v 400 km/h. The aircraft is tracked by radar at O For the position B, deter-mine the values o f i i , 0, and tj>.
za sin 2K nt
Problem 2/178
88 Chapter 2 K i n e m a t i c s of P a r t i c l e s
2 / 1 7 9 The rotating nozzle sprays a large circular' area and turns with the constant angular rate 8 K. Parti-cles of water move along the tube at the constant rate I = c relative to the tube. Write expressions for the magnitudes of the velocity and acceleration of a water particle P for a given position I in the ro-tating tube.
Ans. [/ = Jc2 + K2t2 sin2 j3 a = if sin K2!2 + 4c2
Problem 2/179
2/180 The small block P travels with constant speed V in the circular path of radius r on the inclined surface, if B 0 at time t = 0, determine the x-, y-, and
¿-components of velocity and acceleration as func-tions of time.
Problem 2/180
2/181 At the bottom of a vertical loop in the x-y plane at an altitude of 400 m, the airplane has a speed of 600 km/h with no horizontal A'-acceleration. The radius of curvature of the loop at the bottom is 1200 m. For the radar tracking at O, determine the recorded values of R and <t> for this instant.
An$, R : 34.4 m/s2, ij> = 0.01038 rad/s2
\ y
Problem 2/181
2/182 The robotic device of Prob. 2/162 now rotates about a fixed vertical axis while its arm extends and ele-vates. At a given instant, <j> = 30°, tb 10 deg/s = constant, I = 0.5 m, / = 0.2 m/s, I = —0.3 m/s2, and !i = 20 deg/s = constant. Determine the mag-nitudes of the velocity v and the acceleration a of the gripped part P.
Problem 2/152
Article 2/7 P r o b l e m s 8 9
2/183 The base structure of the flrctruck ladder rotates