Numerical Methods for Engineers
Numerical Methods for Engineers
7th Edition
ISBN: 9780073397924
Author: Steven C. Chapra Dr., Raymond P. Canale
Publisher: McGraw-Hill Education
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Chapter 24, Problem 35P

Perform the same computation as in Sec. 24.4, but use the following equations:

F ( x ) = 1.6 x 0.045 x 2

θ ( x ) = 0.8 + 0.125 x 0.009 x 2 + 0.0002 x 3

Use 4-, 8-, and 16-segment trapezoidal rules to compute the integral.

Expert Solution & Answer
Check Mark
To determine

To calculate: The work done for the given equations of F(x) and θ(x) using 4-, 8-, and 16-segment trapezoidal rule.

F(x)=1.6x0.045x2θ(x)=0.8+0.125x0.009x2+0.0002x3

Answer to Problem 35P

Solution:

The value of W after application of 4-segment Trapezoidal rule is W=58.71675.

The value of W after application of 8-segment Trapezoidal rule is W=64.8995062.

The value of W after application of 16-segment Trapezoidal rule is W=66.419208.

Explanation of Solution

Given Information:

The given expressions are as follows,

F(x)=1.6x0.045x2θ(x)=0.8+0.125x0.009x2+0.0002x3

Work done in integral form (Refer Sec. 24.4)

W=x0xnF(x)dx

If the direction between the force and displacement changes between initial and final position, then the work done is written as,

W=x0xnF(x)cos[θ(x)]dx …… (1)

Here, θ(x) is the angle between force and displacement.

Formula Used:

Multiple Segment Trapezoidal Rule.

In=(ba)f(x0)+2i=1n1f(xi)+f(xn)2(n)

Calculation:

Calculate the work done.

Substitute the value of F(x) and θ(x) in equation (1)

W=030(1.6x0.045x2)cos(0.8+0.125x0.009x2+0.0002x3)dx

Apply 4-Segment Trapezoidal rule.

Calculate the value of x, when n=4,xn=30 and x0=0,

x=xnx0n=3004=7.5

Divide the interval from 0 to 30, and x=7.5,

So, the value of x after each iteration is x=0,7.5,15,22.5 and 30.

Calculate F(0) for x=0,

F(0)=1.6(0)0.045(0)2=0

Calculate θ(0) for x=0,

θ(0)=0.8+0.125(0)0.009(0)2+0.0002(0)3=0.8

Calculate F(0)cos(θ(0)) for x=0,

F(0)cos(θ(0))=0cos(0.8)=0

Calculate F(x) for x=7.5,

F(7.5)=1.6(7.5)0.045(7.5)2=9.46875

Calculate θ(7.5) for x=7.5,

θ(7.5)=0.8+0.125(7.5)0.009(7.5)2+0.0002(7.5)3=1.315625

Calculate F(7.5)cos(θ(7.5)) for x=7.5,

F(7.5)cos(θ(7.5))=9.46875×cos(1.315625)=2.39002

Calculate F(x), for x=15,

F(15)=1.6(15)0.045(15)2=13.875

Calculate θ(15) for x=15,

θ(15)=0.8+0.125(15)0.009(15)2+0.0002(15)3=1.325

Calculate F(15)cos(θ(15)) for x=15,

F(15)cos(θ(15))=13.875×cos(1.325)=3.37619

Similarly, calculate for x=22.5 and 30 thentabulate the values as shown below.

xF(x)θ(x)F(x)cos(θ(x))000.807.59.468751.3156252.390021513.8751.3253.3761822.513.218751.3343753.09616307.51.852.06692

Calculate the solution using Trapezoidal rule,

W=x2(F(0)cos(θ(0))+2F(7.5)cos(θ(7.5))+2F(15)cos(θ(15))+2F(22.5)cos(θ(22.5))+F(30)cos(θ(30)))

Substitute function values from above table for x=7.5.

W=7.52(0+22.39002+23.37618+23.096162.06692)=58.71675

Hence, the value of W after application of 4-segment Trapezoidal rule is W=58.71675.

Apply 8-Segment Trapezoidal rule.

Calculate the value of x, when n=8,xn=30 and x0=0,

x=xnx0n=3008=3.75

Divide the interval from 0 to 30, and x=3.75,

So, the value of x after each iteration is x=0,3.75,7.5,11.25,15,18.75,22.5,26.25 and 30.

Calculate F(0) for x=0,

F(0)=1.6(0)0.045(0)2=0

Calculate θ(0) for x=0,

θ(0)=0.8+0.125(0)0.009(0)2+0.0002(0)3=0.8

Calculate F(0)cos(θ(0)) for x=0,

F(0)cos(θ(0))=0cos(0.8)=0

Calculate F(3.75) for x=3.75,

F(3.75)=1.6(3.75)0.045(3.75)2=5.3671875

Calculate θ(3.75) for x=3.75,

θ(3.75)=0.8+0.125(3.75)0.009(3.75)2+0.0002(3.75)3=1.152734375

Calculate F(3.75)cos(θ(3.75)) for x=3.75,

F(3.75)cos(θ(3.75))=5.3671875×cos(1.152734375)=2.1790249

Calculate F(x) for x=7.5,

F(7.5)=1.6(7.5)0.045(7.5)2=9.46875

Calculate θ(7.5) for x=7.5,

θ(7.5)=0.8+0.125(7.5)0.009(7.5)2+0.0002(7.5)3=1.315625

Calculate F(7.5)cos(θ(7.5)) for x=7.5,

F(7.5)cos(θ(7.5))=9.46875×cos(1.315625)=2.39002

Similarly, calculate x=11.25,15,18.75,22.5,26.25 and 30 then tabulate all solutions as shown below.

xF(x)θ(x)F(x)cos(θ(x))000.803.755.36718751.1527342.179027.59.468751.3156252.3900211.2512.30468751.35195352.671351513.8751.3253.3761818.7514.17968751.29804683.8197322.513.218751.3343753.0961626.2510.99218751.49726560.807535307.51.852.06692

Calculate the solution using Trapezoidal rule,

W=x2(F(0)cos(θ(0))+2F(3.75)cos(θ(3.75))+2F(7.5)cos(θ(7.5))+2F(26.25)cos(θ(26.25))+F(30)cos(θ(30)))

Substitute function values from above table x=3.75.

W=3.752(0+22.17902+22.39002+22.67135+23.37618+23.81973+23.09616+20.8075352.06692)=64.8995062

Hence, the value of W after application of 8-segment Trapezoidal rule is W=64.8995062

Apply 16-Segment Trapezoidal rule.

Calculate the value of x, when n=8,xn=30 and x0=0,

x=xnx0n=30016=1.875

Divide the interval from 0 to 30, and x=1.875,

So, the value of x after each iteration is,

x=(0,1.875,3.75,5.625,7.5,9.375,11.25,13.125,15,16.875,18.75,20.625,22.5,24.375,26.25,28.125 and 30).

Calculate F(0) for x=0,

F(0)=1.6(0)0.045(0)2=0

Calculate θ(0) for x=0,

θ(0)=0.8+0.125(0)0.009(0)2+0.0002(0)3=0.8

Calculate F(0)cos(θ(0)) for x=0,

F(0)cos(θ(0))=0cos(0.8)=0

Calculate F(1.875) for x=1.875,

F(1.875)=1.6(1.875)0.045(1.875)2=2.841796875

Calculate θ(1.875) for x=1.875,

θ(1.875)=0.8+0.125(1.875)0.009(1.875)2+0.0002(1.875)3=1.004052734375

Calculate F(1.875)cos(θ(1.875)) for x=1.875,

F(1.875)cos(θ(1.875))=2.841796875×cos(1.004052734375)=1.525725560

Calculate F(3.75) for x=3.75,

F(3.75)=1.6(3.75)0.045(3.75)2=5.3671875

Calculate θ(3.75) for x=3.75,

θ(3.75)=0.8+0.125(3.75)0.009(3.75)2+0.0002(3.75)3=1.152734375

Calculate F(3.75)cos(θ(3.75)) for x=3.75,

F(3.75)cos(θ(3.75))=5.3671875×cos(1.152734375)=2.1790249

Similarly, calculate x=(5.625,7.5,9.375,11.25,13.125,15,16.875,18.75,20.625,22.5,24.375,26.25,28.125 and 30) then tabulate all the solutions as shown below.

xF(x)θ(x)F(x)cos(θ(x))000.801.8752.84179681.00405271.525723.755.36718751.1527342.179025.6257.57617181.2539552.360487.59.468751.3156252.390029.37511.0449211.3456542.4657211.2512.30468751.35195352.6713513.12513.2480461.3424312.999161513.8751.3253.37618 16.87514.1855461.307568     3.69106     18.7514.17968751.29804683.8197320.62513.8574211.30434573.6487822.513.218751.3343753.0961624.37512.2636711.3960442.1322026.2510.99218751.49726560.80753528.1259.4042961.6459470.706077307.51.852.06692

Calculate the solution using Trapezoidal rule,

W=x2(F(0)cos(θ(0))+2F(1.875)cos(θ(1.875))+2F(3.75)cos(θ(3.75))+2F(28.125)cos(θ(28.125))+F(30)cos(θ(30)))

Substitute function values from above table for x=1.875.

W=1.8752(0+21.52572+22.17902+22.36048+22.39002+22.46572+22.67135+22.99916+23.37618+23.69106+23.81973+23.64878+23.09616+22.13220+20.807535+2(0.706077)2.06692)=66.41920875

Hence, the value of W after application of 16-segment Trapezoidal rule is W=66.419208.

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Chapter 24 Solutions

Numerical Methods for Engineers

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