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
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Chapter 11 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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