calcIIIstudent Group Title 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 2 years ago

1. calcIIIstudent Group Title

2. calcIIIstudent Group Title

I'm completely lost on this one, I don't even know what formula to use.

3. experimentX Group Title

|dw:1344456883598:dw|

4. experimentX Group Title

|dw:1344456975045:dw| change everything into t's ... since parameter is given, it should be easy to find it.

5. experimentX Group Title

the 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.

6. calcIIIstudent Group Title

Well, 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

7. experimentX Group Title

for a conservative field, |dw:1344457305326:dw| isn't this definition of conservative field?

8. experimentX Group Title

|dw:1344457452297:dw| moreover F = -div V

9. colorful Group Title

$\int\limits_C\vec F\cdot d\vec r=\int_a^b\vec F(\vec r(t))\cdot\vec r'(t)dt$I don't see where you need $$\nabla f$$ in this problem...

10. colorful Group Title

on nvm

11. colorful Group Title

so this is going to involve the fundamental theorem for line integrals

12. calcIIIstudent Group Title

Also this is a friend of mines test result (I'm studying for finals) and I have no idea what she did.

13. colorful Group Title

$\int\limits_C\nabla f\cdot d\vec r=f(\vec r(b))-f(\vec r(a))$

14. experimentX Group Title

she did what i suggested ... first find the curl of field (to make sure that it is conservative) though it isn't necessary.

15. experimentX Group Title

if 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 ...

16. calcIIIstudent Group Title

If finding the curl wasn't necessary, then where did she get these numbers from?

17. experimentX Group Title

|dw:1344458019983:dw|

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|dw:1344458032365:dw|

19. experimentX Group Title

|dw:1344458052253:dw| use those values of t's ... and find the x coordinate and y coordinate. at t=0 and t=pi

20. colorful Group Title

Work 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$

21. colorful Group Title

so$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$

22. colorful Group Title

questions?

23. experimentX Group Title

this way is super easy ... if you are preparing for finals ... try to work on both methods.

24. colorful Group Title

25. colorful Group Title

btw 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:$W=\int\limits_C\vec F\cdot d\vec r=\int\limits_C\nabla f\cdot d\vec r=f(\vec r(b))-f(\vec r(a))$

26. calcIIIstudent Group Title

Sorry, my internet crashed. But yeah, that pretty much summed up all my questions. Thanks!

27. experimentX Group Title

yw