Concept explainers
Repeat Prob. 24.35, but use (a) Simpson's
(a)
To calculate: The work done for the given equations of
Answer to Problem 36P
Solution:
The work done byusing 4-segment Simpson 1/3 rule is
The work done by using 8-segment Simpson 1/3 rule is
The work done by using 16-segment Simpson 1/3 rule is
Explanation of Solution
Given Information:
The given expressions are as follows,
Work done in integral form (Refer Sec. 24.4)
If the direction between the force and displacement changes between initial and final position, then the work done is written as,
Here,
Formula Used:
Simpson’s 1/3 rule.
Calculation:
Calculate the work done.
Substitute the value of
Force
And,
4-segment Simpson’s 1/3 rule.
Calculate integral from
0 | 0 |
1 | 7.5 |
2 | 15 |
3 | 22.5 |
4 | 30 |
Calculate
Calculate
Similarly, calculate
Calculate
Calculate
Calculate
Similarly, calculate
Calculate
Substitute value of
Calculate
Substitute value of
Similarly, calculate all the other values.
All the values which are calculated in are tabulated below,
0 | 0 | 0 | 0.8 | 0 |
1 | 7.5 | 9.46875 | 1.315625 | 2.39001847 |
2 | 15 | 13.875 | 1.325 | 3.376187019 |
3 | 22.5 | 13.21875 | 1.334375 | 3.096161858 |
4 | 30 | 7.5 | 1.85 | –2.066926851 |
Apply Simpson’s 1/3 rule to calculate work.
According to Simpson’s 1/3 rule integral
Here,
Work done is given as,
Here,
Calculate
Substitute values of
Hence, the value of integral is
8-segment Simpson’s 1/3 rule.
For eight segmented rule value of
All the values are tabulated below.
0 | 0 | 0 | 0.8 | 0 |
1 | 3.75 | 5.367188 | 1.152734 | 2.179025 |
2 | 7.5 | 9.46875 | 1.315625 | 2.390018 |
3 | 11.25 | 12.30469 | 1.351953 | 2.671355 |
4 | 15 | 13.875 | 1.325 | 3.376187 |
5 | 18.75 | 14.17969 | 1.298047 | 3.819728 |
6 | 22.5 | 13.21875 | 1.334375 | 3.096162 |
7 | 26.25 | 10.99219 | 1.497266 | 0.807535 |
8 | 30 | 7.5 | 1.85 | –2.06693 |
Apply Simpson’s 1/3 rule to calculate work done,
According to Simpson’s 1/3 rule integral
Work done is given by,
Here,
Substitute values of
Calculate
Hence, the value of integral is
16 segment Simpson’s 1/3 rule.
The whole interval from
All the values are tabulated below.
0 | 0 | 0 | 0.8 | 0 |
1 | 1.875 | 2.841797 | 1.004053 | 1.525726 |
2 | 3.75 | 5.367188 | 1.152734 | 2.179025 |
3 | 5.625 | 7.576172 | 1.253955 | 2.360482 |
4 | 7.5 | 9.46875 | 1.315625 | 2.390018 |
5 | 9.375 | 11.04492 | 1.345654 | 2.465721 |
6 | 11.25 | 12.30469 | 1.351953 | 2.671355 |
7 | 13.125 | 13.24805 | 1.342432 | 2.999159 |
8 | 15 | 13.875 | 1.325 | 3.376187 |
9 | 16.875 | 14.18555 | 1.307568 | 3.691061 |
10 | 18.75 | 14.17969 | 1.298047 | 3.819728 |
11 | 20.625 | 13.85742 | 1.304346 | 3.648784 |
12 | 22.5 | 13.21875 | 1.334375 | 3.096162 |
13 | 24.375 | 12.26367 | 1.396045 | 2.132203 |
14 | 26.25 | 10.99219 | 1.497266 | 0.807535 |
15 | 28.125 | 9.404297 | 1.645947 | –0.70608 |
16 | 30 | 7.5 | 1.85 | –2.06693 |
Apply Simpson’s 1/3 rule to calculate work done.
According to Simpson’s 1/3 rule integral
In the above expression
Work done is given by,
Here,
Substitute values of
Calculate
Hence, the work done is
(b)
To calculate: The work done for the given equations of
Answer to Problem 36P
Solution: The work done is obtained to be
Explanation of Solution
Given Information:
The given expressions are as follows,
Work done in integral form (Refer Sec. 24.4).
If the direction between the force and displacement changes between initial and final position, then the work done is written as,
Here,
Formula Used:
Single segment trapezoidal rule.
Multiple application trapezoidal rule.
An estimate of relative percentage error.
Calculation:
The integral is,
And,
Here,
For Romberg Iteration- 1, 2, 4 and 8 segment trapezoidal rule integral needs to be calculated which will be used for complexity calculation in higher order correction of integral estimates.
The Integral for single segment trapezoidal rule is,
Substitute
The Integral for multiple application trapezoidal rules is,
Here,
For calculation of
Hence,
So,
For calculation of
Hence,
So,
For calculation of
So,
Hence,
The complexity notation for Romberg iteration is,
An estimate of relative percentage error is,
Setting up the table for Romberg iteration,
The first table of
From the Romberg table of iteration, the work done is obtained to be
(c)
To calculate: The work done for the given equations of
Answer to Problem 36P
Solution: The work done using Gauss Quadrature is obtained to be
Explanation of Solution
Given Information:
The given expressions are as follows,
Work done in integral form (Refer Sec. 24.4)
If the direction between the force and displacement changes between initial and final position, then the work done is written as,
Here,
Formula Used:
Change of variables formula,
Gauss Quadrature formula for integral calculation,
Calculation:
The integral is,
And,
Change of variables is required so as to transform the original limits of given original integral to
Substitute
Differentiate
So, the function is,
Substitute the value of
The integral becomes,
Therefore, Integrals is suitable for Gauss Quadrature calculation.
The weighting factors for 2-point Gauss Quadrature evaluation is,
And,
The Integral is given by,
Substitute the values from above.
Three-Point Gauss Quadrature factor will be tried out to reduce the truncation error.
The weighting factors for 3-Point Gauss Quadrature evaluation is,
And,
The Integral is given by,
Substitute the value from above.
The obtained value is acceptable as the relative percentage error is less than1%.
Hence, the value of
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