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Nairaland Forum / Nairaland / General / Education / Nairaland Mathematicians And Physicists, Come And Try This. (1696 Views)
Mathematicians In The House come over Here / Nairaland Mathematicians, Please Come And Help Me Solve This / Mathematicians And Physicists. Your Help Needed (2) (3) (4)
Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 11:00am On Dec 10, 2019 |
1) If a continuously varying force F(x) moves an object on the xy-plane through arc (whose derivative is continous) given by g(x) from x=a to x=b (on the interval [a, b]). Derive a formula for the work done by the force in moving the objects from x=a to x=b along the arc. 2) If the force is F(x) = 2x-3 and the arc is g(x) = ln x, find the work done if an object is moved from x= 1 to x= 10 ![]() ![]() ![]() Cc. Mrshape, Darivie04, MathsEconomics, and others. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Mrshape: 5:50pm On Dec 10, 2019 |
Martinez39:Hmmmm. I will attempt later sir |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by frankdoz: 10:19pm On Dec 10, 2019 |
Very simple question, how much will you pay me to solve it? 1 Like 1 Share |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 3:36am On Dec 13, 2019 |
So no one is up to the task? |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 10:51am On Dec 13, 2019 |
Martinez39:Should I post the answer? |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by frankdoz: 2:54pm On Dec 13, 2019 |
Martinez39:I have solved it! |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 3:51pm On Dec 13, 2019 |
frankdoz:Post your answer. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Nobody: 9:17am On Dec 17, 2019 |
Martinez39:Really cool problem, wish I had seen it earlier. I might drop a solution later today. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by frankdoz: 10:08am On Dec 17, 2019 |
OK, check my answer! |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 10:17am On Dec 17, 2019 |
Darivie04:Okay then. I will drop my solution tomorrow morning. I await your solution. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 10:26am On Dec 17, 2019 |
frankdoz:Wrong! Try again. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by frankdoz: 10:34am On Dec 17, 2019 |
Martinez39:then drop your answer, stop wasting time! |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Nobody: 11:31am On Dec 17, 2019 |
Are you sure F(x) is not equal to 2x^3? |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 11:32am On Dec 17, 2019 |
frankdoz:I wanted to drop the solution this afternoon but Darivie04 said he didn't see the question until today and he would be dropping his solution later today. For his sake, I will withhold my answer until 8:00pm today. Though I wanted to wait till tomorrow but I will unfailingly release my answer by 8:00pm since you are eager to see it. But if Darivie04 drops a solution now (wrong or correct), I will release my answer soon after his. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 11:35am On Dec 17, 2019 |
Darivie04:F(x) = 2x-3 |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Nobody: 12:25pm On Dec 17, 2019 |
. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 12:29pm On Dec 17, 2019 |
Darivie04: ![]() |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Nobody: 12:33pm On Dec 17, 2019 |
Martinez39:Wait abeg ![]() |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Nobody: 12:39pm On Dec 17, 2019 |
2.Work Done = 278.3314206 Joules |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 1:27pm On Dec 17, 2019 |
MY SOLUTION — QUESTION ONE ANSWER Partition the interval [a, b] in n subinterval of width Δxi. Δxi is the width of the i -th subinterval. The work done in the i -th subinterval is approximated by F(ci) [ Δxi² + Δyi² ]½ —— (1) (force times distance) where ci is any value of x in the i -th subinterval and Δyi is the total change in y in the i -th subinterval) By rewriting, (1) takes the form F(ci)[ 1 + (Δyi/Δxi)² ]½ Δxi From the Mean Value theorem we know that since g(x) is continous on the interval [a, b] and differentiable on the interval (a, b), then on each subinterval, there exists an x value, say di, for which the derivative is equal to the average rate over the subinterval ie. g'(di) = Δyi/Δxi (for the i -th subinterval) So we can write F(ci) [ 1 + (g'(di))² ]½ Δxi By summing the approximations on each subinterval, we can approximate the work done W by the force F(x) on moving an object on the arc g(x) by W ≈ Σ F(ci)[ 1 + (g'(di))² ]½ Δxi By taking limit of this Riemann sum as the norm of the sum (||Δ||) approaches zero, we get the exact work done. The means that the work done is The integral of F(x) [ 1 + (g'(x))² ]½ from x = a to x = b Cc. Darivie04, frankoz |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Nobody: 1:45pm On Dec 17, 2019 |
Darivie04:Have you seen this @Martinez39? |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Nobody: 1:49pm On Dec 17, 2019 |
Martinez39:I was able to get the question up to this part. It was that force function in no.2 that confused me for awhile. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 1:50pm On Dec 17, 2019 |
Darivie04:I have seen it but it is incorrect. Evaluate the definite integral I gave and compare the answers. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Nobody: 2:16pm On Dec 17, 2019 |
Martinez39:I have seen my error in the second solution I posted. I also saw that our formulas are actually different so I'll just write down my full solution and put it here. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 2:25pm On Dec 17, 2019 |
Darivie04:Did your answer accord with the definite integral I gave? |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Nobody: 3:06pm On Dec 17, 2019 |
Martinez39:We got different integrals for number 1. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Nobody: 3:09pm On Dec 17, 2019 |
My solution is based on the fact that the work done is the area beneath the force distance curve. My xf is gotten from the formula of an arc length. If I'm making a mistake somewhere I'll be glad to see it.
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Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 3:18pm On Dec 17, 2019 |
Darivie04:Not correct. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Nobody: 3:46pm On Dec 17, 2019 |
Martinez39:Why is it not correct? |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 4:25pm On Dec 17, 2019 |
Darivie04:Your integral is an improper integral because F(x) = 2x-3 has an infinite discontinuity at x = 0. Moreover, your integral diverges. I guess that was why you got Work Done = ∞. ![]() |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Nobody: 4:38pm On Dec 17, 2019 |
Martinez39:But if you use that same force function for a simpler problem say raising an object 20m or any other simple Work related problem. You would have to use a similar integral(with the only difference being the upper bound) and you would still encounter the same problems. E.g A variable force F(x)=2x-3 acts on an object and displaces the object by 20m. Calculate the work done. |
Re: Nairaland Mathematicians And Physicists, Come And Try This. by Martinez39(m): 4:44pm On Dec 17, 2019 |
Darivie04:True. The reason is because the force function has an infinite discontinuity at x = 0. Any definite integral of the force function with x = 0 as the lower limit of integration is an improper integral that diverges. ![]() That is why I used x = 1 as the lower limit of integration. ![]() 1 Like 1 Share |
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