Concept explainers
Some parasailing systems use a winch to pull the rider back to the boat. During the interval when θ is between 20° and 40° (where t = 0 at θ = 20°), the angle increases at the constant rate of 2°/s. During this time, the length of the rope is defined by the relationship
Fig. P11.163 and P11.164
(a)
Plot the magnitude of the velocity of the parasailer as a function of time.
Explanation of Solution
Given Information:
During the interval the
The angle
The length of the rope is define by the relationship (r) of
The boat is travelling at a constant velocity
Calculation:
Convert the knot to feet per second.
Consider
Show the Free body diagram of parasailer and boat as in Figure (1).
Write the velocity
The acceleration vector of the boat is as follows:
Differentiate angle
Differentiate radius (r) with respective to time (t).
Differentiate
Write the expression for velocity vector
Here,
Write the expression for acceleration vector
Here,
Write the velocity vector
Write the acceleration vector
Write the velocity vector
Substitute
Calculate velocity vector of parasailer at an angle
Substitute 0 for t,
Here,
Calculate the velocity
Substitute
The time (t) is increase with 1 sec for an angle of
Similarly, calculate the velocity
Summarize the calculated values of velocity as in Table (1).
Time(t) (sec) | Radius (r) | |||||
0 | 20 | 600.000 | 0.000 | 32.463 | 19.681 | 37.963 |
1 | 22 | 599.875 | -0.313 | 33.434 | 19.298 | 38.603 |
2 | 24 | 599.293 | -0.884 | 34.616 | 18.751 | 39.369 |
3 | 26 | 598.051 | -1.624 | 35.911 | 18.051 | 40.193 |
4 | 28 | 596.000 | -2.500 | 37.274 | 17.195 | 41.050 |
5 | 30 | 593.012 | -3.494 | 38.676 | 16.180 | 41.924 |
6 | 32 | 588.977 | -4.593 | 40.090 | 15.001 | 42.804 |
7 | 34 | 583.795 | -5.788 | 41.494 | 13.658 | 43.684 |
8 | 36 | 577.373 | -7.071 | 42.867 | 12.149 | 44.555 |
9 | 38 | 569.625 | -8.438 | 44.190 | 10.474 | 45.415 |
10 | 40 | 560.472 | -9.882 | 45.446 | 8.635 | 46.259 |
Plot the magnitude of the velocity of the parasailer as a function of time as in Figure (1).
(a)
The magnitude of the acceleration
Answer to Problem 11.164P
The magnitude of the acceleration
Explanation of Solution
Given Information:
During the interval the
The angle
The length of the rope is define by the relationship (r) of
The boat is travelling at a constant velocity
Calculation:
Write the expression for acceleration vector
Substitute
Calculate the acceleration vector
Substitute 5 sec for t,
Here,
Calculate the acceleration
Substitute
Therefore, the magnitude of the acceleration
Want to see more full solutions like this?
Chapter 11 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
- 4. An airplane is travelling horizontally at an altitude of 1768.01m. It released a bomb targeting an object 1500m from its release point. Determine the velocity of the plane and the time when the bomb reached its target. Sign convention +y +x 40° 1500marrow_forwardAircraft A is flying horizontally at an altitude of 10.6 km and is increasing its speed at the rate of 2 m/s each second. Aircraft B, flying in the same vertical plane at an altitude of 16 km, has a constant speed of 1400 km/h. If A has a speed of 1100 km/h at the instant when 0 = 30°, determine the values of 7 and 0 for this same instant. (* = 12.9 m/s², 0 = −0.0037 rad/s²) A 0 Barrow_forward2- The rotation of rod OA about O is defined by the relation 0 = n(4t² – 8t), where 0 and t are expressed in radians and seconds, respectively. Collar B slides along the rod so that its distance from 0 is r = 10 + 6 sin at, where r and t are expressed in inches and seconds, respectively. When t = 1 s, determine (a) the velocity of the collar, (b) the total acceleration of the collar, (c) the acceleration of the collar relative to the rod.arrow_forward
- Two airplanes are performing at an air show. Plane A is travelling along a straight inclined path (0 = 41°) and has a constant velocity of va = 136 m. Plane B enters the circular path (r = 135 meters) at point O with a velocity of (vB)0o = 46 m and reaches to the velocity of vB = 63 m when it has travelled 30° around the circular path (the instant shown on the picture). Determine the magnitude of the normal component of acceleration vector of plane B (in m) at this instant. A 30% Вarrow_forwardA Scotch yoke is a mechanism that transforms the circular motion of a crank into the reciprocating motion of a shaft (or vice versa). It has been used in a number of different internal combustion engines and in control valves. In the Scotch yoke shown, the acceleration of Point A is defined by the relation a=-1.5sin(kt) , where a and t are expressed in m/s2 and seconds, respectively, and k=3 rad/s. Knowing that x=0 and v=0.6 m/s when t =0, determine the position of Point A when t=0.5 s.arrow_forwardTo study the performance of a race car, a high-speed camera is positioned at point A. The camera is mounted on a mechanism which permits it to record the motion of the car as the car travels on straightaway BC. It took 0.5 s for the car to travel from the position θ = 60° to the position θ = 35°. Knowing that b = 24 m, determine the average speed of the car during the 0.5-s interval. The average speed of the car during the 0.5-s interval is ______ km/h.arrow_forward
- A loaded railroad car is rolling at a constant velocity when it couples with a spring and dashpot bumper system. After the coupling, the motion of the car is defined by the relation 4.8 60 sin16 t xe t − = where x and t are expressed in mm and seconds, respectively. Determine the position, the velocity and the acceleration of the railroad car when (a) t = 0, (b) t = 0.3 s.arrow_forward3. The spatial motion of a particle is described by X = 3t 2 + 4t y = -4t 2 + 3t Z = -6t +9 where the coordinates are measured in feet and the time t is in seconds. (a) Determine the velocity and acceleration vectors of the particle as functions of time. (b) Verify that the particle is undergoing plane motion (the motion is not in a coordinate plane) by showing that the unit vector perpendicular to the plane formed by v and a is constant.arrow_forwardA loaded railroad car is rolling at a constant velocity when it couples with a spring and dashpot bumper system. After the coupling, the motion of the car is defined by the relation x = 60e - 4.8t sin 16t, where x and t are expressed in millimeters and seconds, respectively. Determine the position, the velocity, and the acceleration of the railroad car when (a) t= 0, (b) t = 0.3 s.arrow_forward
- A horse is used to help stack hay in a barn. At the instant shown, the length of rope, r, is 7 m, the angle the rope makes with the horizontal is = 50°. The 750-kg horse is walking at a velocity of 1.5 m/s and is speeding up at a rate of 0.25 m/s². The mass of the bale is 60 kg. Determine at this instant, (a) the speed of the bale, (b) the tension in the rope (c) the average force between the ground and the horse's feet.arrow_forwardThe 0.5-lb projectile A is subjected to a drag force of magnitude kv2, where the constant k = 0.0003 lb-sec²/ft2. This drag force always opposes the velocity v. At the instant depicted, v = 210 ft/sec, 0 = 41°, and r = 495 ft. Determine the corresponding values of r and 0. y | ku² Answers: i M. 0 = i 0 T A V 170 --x ft/sec² rad/sec²arrow_forward3. A motorcycle patrolman starts from rest at A 2 sec. after a car speeding at the constant rate of 120 kph passes point A. If the patrolman accelerates at the rate of 6 m/s? until he reaches his maximum permissible speed of 150 kph, which he maintains. a. Find the time for the patrolman to reach a maximum permissible speed of 150 kph. b. How long did it take the patrolman to overtake the car from the time he starts from A.? c. Calculate the distance S from point A to the point at which he overtakes the car.arrow_forward
- Elements Of ElectromagneticsMechanical EngineeringISBN:9780190698614Author:Sadiku, Matthew N. O.Publisher:Oxford University PressMechanics of Materials (10th Edition)Mechanical EngineeringISBN:9780134319650Author:Russell C. HibbelerPublisher:PEARSONThermodynamics: An Engineering ApproachMechanical EngineeringISBN:9781259822674Author:Yunus A. Cengel Dr., Michael A. BolesPublisher:McGraw-Hill Education
- Control Systems EngineeringMechanical EngineeringISBN:9781118170519Author:Norman S. NisePublisher:WILEYMechanics of Materials (MindTap Course List)Mechanical EngineeringISBN:9781337093347Author:Barry J. Goodno, James M. GerePublisher:Cengage LearningEngineering Mechanics: StaticsMechanical EngineeringISBN:9781118807330Author:James L. Meriam, L. G. Kraige, J. N. BoltonPublisher:WILEY