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 2 years ago
Calc III:
Determine how much work was done by an object in moving along an arc of the cycloid given by:
Image attached
 2 years ago
Calc III: Determine how much work was done by an object in moving along an arc of the cycloid given by: Image attached

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calcIIIstudent
 2 years ago
Best ResponseYou've already chosen the best response.0I'm completely lost on this one, I don't even know what formula to use.

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1344456883598:dw

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1344456975045:dw change everything into t's ... since parameter is given, it should be easy to find it.

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1the other method, ... check if the field is conservative or not, if the field is conservative, the difference in the potential should give you the work done.

calcIIIstudent
 2 years ago
Best ResponseYou've already chosen the best response.0Well, the field is conservative, and the parameter is 0 to pi, but what I'm not understanding is what to do with the F*dr

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1for a conservative field, dw:1344457305326:dw isn't this definition of conservative field?

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1344457452297:dw moreover F = div V

colorful
 2 years ago
Best ResponseYou've already chosen the best response.2\[\int\limits_C\vec F\cdot d\vec r=\int_a^b\vec F(\vec r(t))\cdot\vec r'(t)dt\]I don't see where you need \(\nabla f\) in this problem...

colorful
 2 years ago
Best ResponseYou've already chosen the best response.2so this is going to involve the fundamental theorem for line integrals

calcIIIstudent
 2 years ago
Best ResponseYou've already chosen the best response.0Also this is a friend of mines test result (I'm studying for finals) and I have no idea what she did.

colorful
 2 years ago
Best ResponseYou've already chosen the best response.2\[\int\limits_C\nabla f\cdot d\vec r=f(\vec r(b))f(\vec r(a))\]

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1she did what i suggested ... first find the curl of field (to make sure that it is conservative) though it isn't necessary.

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1if the field is conservative, and you know the potential ... the line integral of Field along the path (or so called work done in physics) = change in potential ...

calcIIIstudent
 2 years ago
Best ResponseYou've already chosen the best response.0If finding the curl wasn't necessary, then where did she get these numbers from?

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1344458019983:dw

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1344458032365:dw

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1344458052253:dw use those values of t's ... and find the x coordinate and y coordinate. at t=0 and t=pi

colorful
 2 years ago
Best ResponseYou've already chosen the best response.2Work is defined as\[W=\int\limits_C\vec F\cdot d\vec r=\int\limits_C\nabla f\cdot d\vec r=f(\vec r(b))f(\vec r(a))\]the potential function for the conservative vector field is\[f=\frac13x^3+\frac13y^3\]and the line is\[\vec r(t)=\langle3(t\sin t),3(1\cos t),\rangle\]we have\[\vec r(\pi)=\langle3\pi,6\rangle\]\[\vec r(0)=\langle0,0\rangle\]

colorful
 2 years ago
Best ResponseYou've already chosen the best response.2so\[W=\int\limits_C\vec F\cdot d\vec r=\int\limits_C\nabla f\cdot d\vec r=f(\vec r(\pi))f(\vec r(0))\]\[\left(\frac13(3\pi)^3+\frac136^3\right)0=9\pi^3+72\]

experimentX
 2 years ago
Best ResponseYou've already chosen the best response.1this way is super easy ... if you are preparing for finals ... try to work on both methods.

colorful
 2 years ago
Best ResponseYou've already chosen the best response.2yes, always learn more than one way when possible !

colorful
 2 years ago
Best ResponseYou've already chosen the best response.2btw when I said "work is defined as" I only meant the first part of the expression\[W=\int\limits_C\vec F\cdot d\vec r\]the next part only follows if the force \(\vec F\) is conservative:\[W=\int\limits_C\vec F\cdot d\vec r=\int\limits_C\nabla f\cdot d\vec r=f(\vec r(b))f(\vec r(a))\]

calcIIIstudent
 2 years ago
Best ResponseYou've already chosen the best response.0Sorry, my internet crashed. But yeah, that pretty much summed up all my questions. Thanks!
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