REVIEW PROBLEMS
Article 2 1 0 Review Problems 11 ï
2 / 2 3 9 The vertically-fired rocket and tracking radar' of Prob. 2/147 are shown again here. At the instant when 0 = 60s, measurements give 8 = 0.03 rad/sec and r 25,000 ft, and the vertical acceleration of the rocket is found to be a = 64 ft/sec2. For this in-stant determine the values of ? and 8,
Am. r = 77.9 ft/sec2
S = -1.838(10 3) rad/sec2
Problem 2/239
2/240 The vertical displacement of cylinder A in meters is given by y = t2<(4 where t is in seconds. Calculate the downward acceleration aB of cylinder B. Iden-tify the number of degrees of freedom.
W J J
2/241 A jet aircraft pulls up into a vertical curve as shown. As it passes the position where 0 - 30°, its speed is 1000 km/h and is decreasing at the rate of 15 km/h per second. If the radius of curvature p of the flight path is 1.5 km at this point, calculate the corresponding horizontal and vertical components, x and y, of the acceleration of the aircraft.
Ans. x = - 2 9 . 3 m/s2 y 42.5 m/s2
C\
P
Probiem 2/241
2 / 2 4 2 The launching catapult of the aircraft carrier gives the jet fighter a constant acceleration of 50 m/s2
from rest relative to the flight deck and launches the aircraft in a distance of 100 m measured along the angled takeoff ramp. If the carrier is moving at a steady 30 knots 11 knot = 1.852 km/h), determine the magnitude p of the actual velocity of the fighter when it is launched.
Problem 2/242
Problem 2/240
114 Chapter 2 K i n e m a t i c s of P a r t i c l e s
2/243 Car A negotiates a eui"ve of 60-m radius at a con-stant speed of 50 km/h. When A passes the position shown, car B is 30 m from the intersection and is accelerating south toward the intersection at the rate of 1.5 m / s D e t e r m i n e the acceleration which A appears to have when observed by an occupant of B at this instant.
Arcs. aA j B 4.58 m / s , fi 20.0° west of north
2 / 2 4 5 Cylinder A has a constant downward speed of 1 m/s. Compute the velocity of cylinder B for (o) fi 45°, (6) 8 = 30°, and id 0 = 15°. The spring is in tension throughout the motion range of interest, and the pulleys are connected by the cable of fixed length.
Ans. (a) vB = 0.293 m/s
(6) v
B= 0
(c) vB = - 0 . 2 5 0 m/s
2 / 2 4 4 At the instant depicted, assume that the particle P, which moves on a curved path, is 80 m from the pole O and has the velocity v and acceleration a as indicated. Determine the instantaneous values of T, r, 9, i), the n- and f-components of acceleration, and the radius of curvature f>.
Problem 2/244
I m/s
Problem 2/245
2 / 2 4 6 A particle has the following position, velocity, and acceleration components: x 50 ft, y 25 ft, x :
— 10 ft/sec, y = 10 ft/sec, x - 1 0 ft/sec', and y = 5 ft/sec2. Determine the following quantities: v, a, e„ e,,., a„ a„ a„, a„, p, e „ e„, vn v „ vg, v„, a,, a,, a„, a„, i\ r, t, 0, 0, and f). Express all vectors in terms of i and j, and graph all vectors on one set of x-y axes as you proceed.
2/247 Just after being strack by the club, a golf ball has a velocity of 125 ft/sec directed at 35° to the horizon-tal as shown. Determine the location of the point of impact.
Arts. R = 152.0 yd
125 ft/sec
p 30 yd -I 100 yd -r 25 yd j- • 25 yd -J
Problem 2/247
Article 2 / 1 0 Review P r o b l e m s 1 1 5
2 / 2 4 8 A rocket fired vertically up from the north pole achieves a velocity of 27 000 km/h at an altitude of 350 km when its fuel is exhausted. Calculate the additional vertical height h reached by the rocket before it starts its descent back to the earth. The coasting phase of its flight occurs above the atmos-phere. Consult Fig. 1/1 in choosing the appropriate value of gravitational acceleration and use the mean radius of the earth from Table D/2. {Note:
Launching from the earth's pole avoids considering the effect of the earth's rotation.)
2 / 2 4 9 In the differential pulley hoist shown, the two upper pulleys are fastened together to form an in-tegral unit. The cable is wrapped around the smaller pulley with its end secured to the pulley so that it cannot slip. Determine the upward accelera-tion oo of cylinder B if cylinder A has a downward acceleration of 2 ft/sec2. {Suggestion: Analyze geo-metrically the consequences of a differential move-ment of cylinder A.)
Ans. aB = 0.25 ft/sec2
Problem 2/249
• *Computer-Oriented Problems
*2/2S0 A baseball is dropped from an altitude h = 200 ft and is found to be traveling at 85 ft/sec when it strikes the ground. In addition to gravitational ac-celeration, which may be assumed constant, air resistance causes a deceleration component of magnitude kv , where v is the speed and k is a constant. Determine the value of the coefficient k.
Plot the speed of the baseball as a function of alti-tude y. If the baseball were dropped from a high altitude, but one at which g may still be assumed constant, what would be the terminal velocity vp.
(The terminal velocity is that speed at which the acceleration of gravity and that due to air resis-tance are equal and opposite, so that the baseball drops at a constant speed.) If the baseball were dropped from h = 200 ft, at what speed v' would it strike the ground if air resistance were neglected?
*2/251 At time t 0, the 1.8-lb particle P is given an initial velocity 1/0=1 ft/sec at the position 0 - 0 and sub-sequently slides along the circular path of radius r = 1.5 ft. Because of the viscous fluid and the ef-fect of gravitational acceleration, the tangential acceleration is at g cos ti - ~ v, where the con-stant It 0.2 lb-sec/ft is a drag parameter. Deter-mine and plot both ft and II as functions of the time t over the range 0 i i £ 5 sec. Determine the maximum values of 8 and 8 and the correspond-ing values of t. Also determine the first time at which 9 = 90°.
116 Chapter 2 K i n e m a t i c s of P a r t i c l e s
*2/252 If aE frictional effects are neglected, the expression for the angular acceleration of the simple pcndu-lum is 6 = — cos 6, where g is the acceleration of 8
gravity and 1 is the length of the rod OA. If the pen-dulum has a clockwise angular velocity 8 2 rad's when 0 = 0 at t = 0, determine the time t' at which the pendulum passes the vertical position 8 ~ 90°.
The pendulum length is / = 0.6 m. Also plot the time t versus the angle 0.
*2/253 A ship with a total displacement of 16 000 metric tons (1 metric ton = 1000 kg) starts from rest in stiE water under a constant propeller thrust T -250 kN. The ship develops a total resistance to mo-tion through the water given by R = 4.50f2, where R is in kilonewtons and v is in meters per second.
The acceleration of the ship is a = (T — R)/m, where m equals the mass of the ship in metric tons. Plot the speed v of the ship in knots as a function of the distance s in nautical miles which the ship goes for the first 5 nautical miles from rest. Find the speed after the ship has gone 1 nau-tical mile. What is the maximum speed which the ship can reach?
A/is.
mi " 11.00 knots d , . „ = 14.49 knots
*2/254 By means of the control unit M, the pendulum OA is given an oscillatory motion about the vertical given by 0 = 80 sin j— t, where 0lg O is the maximum
V i
angular displacement in radians, g is the accelera-tion of gravity, / is the pendulum length, and t is the time in seconds measured from an instant when OA is vertical. Determine and plot the mag-nitude a of the acceleration of A as a function of time and as a function of ft over the first quarter cycle of motion. Determine the minimum and maximum values of a and the corresponding val-ues of t and 0. Use the valval-ues i90 = jr/3 radians, I = 0.8 m, and g 9.81 m/s2. (Note: The prescribed motion is not precisely that of a freely swinging pendulum for large amplitudes.!
*2/255 The acceleration of the drag racer is modeled by a = C! — c2v2, where the u2-term accounts for aero-dynamic drag and where C! and c2 are positive constants. I f c i is known to be 30 ft/sec2, determine c2 if the racer completes the |-mi run in 9.4 sec.
Then plot the velocity and displacement as func-tions of time. A drag race is a j - mi straight run from a standing start.
Arcs. ca = 9.28(10 ft 1
1320'
P r o b l e m 2/152