Calculating an Intergral over C Using Force F and Circle C

  • Thread starter jlmac2001
  • Start date
In summary, the task is to find the work done by a force F=yi + xj in going all the way counterclockwise around a circle C given by the equation x^2+y^2+2x=0, using the easiest technique known. The easiest way to approach this is by using Green's theorem, since the force field is "conservative" and its integral around any closed path is 0. The integrand would be d(x)/dx- d(y)/dy = 1-1 = 0. The path C is already given, so there is no need to "get" it. If the problem were to integrate a different function, the integral would be 2 times the area of
  • #1
jlmac2001
75
0
Find the work (intergral over C )F dot dr done by a force F=yi + xj in going all the wy counterclockwise around circle C give by x^2+y^2+2x=0, by the easiet technique you know.

Would i get a double intergral over C (-1) dxdy? How would I get C?
 
Physics news on Phys.org
  • #2
U can interchange dy into dx and vice versa from the equation of circle and it will be easy to integrate
 
  • #3
The easiest way is this: Since d(y)/y= 1= d(x)/dx, this is a "conservative" force field (mathematically, ydx+ xdy is an "exact differential") and so its integral around any closed path is 0.

I'm not sure where you got "-1" from. Using Green's theorem the integrand would be d(x)/dx- d(y)/dy= 1- 1= 0 just as above.

Saying "How would I get C?" makes it sound as if you think C is a constant. You don't have to "get" C: C is the path given.

IF the problem were to integrate, say, ydx+ 3xdy, then we would integrate [tex]\int (\frac{d(3x)}{dx}-\frac{d(y)}{dy})dA [/tex]
= [tex]2\int dA[/tex] which is just 2 times the area of the circle.
 
Last edited by a moderator:

1. How do I calculate an integral over C using force F and circle C?

To calculate an integral over C, you will need to use the formula ∫F(x)dx = ∫F(t)dt, where F(x) represents the force and dx represents the distance. You will also need to use the circle equation x² + y² = r² to represent the circle C. From there, you can solve for the integral by plugging in the necessary values.

2. What is the significance of force F and circle C in calculating an integral?

Force F represents the magnitude and direction of the force acting on an object, while circle C represents the path that the object follows. Together, they help determine the work done on the object, which is represented by the integral.

3. Can I use any circle C and force F to calculate an integral over C?

No, the circle C and force F must be related to each other in order to accurately calculate the integral. For example, if the force is acting tangentially to the circle, then the circle's radius must be used in the calculations. It is important to consider the relationship between the two when choosing which values to use.

4. What is the difference between calculating an integral over C and just calculating an integral?

Calculating an integral over C specifically involves using a circle C in the calculation. This is typically done when the force and path of the object are circular in nature. However, an integral can be calculated for any function, not just those involving circles.

5. Are there any limitations to using force F and circle C in calculating an integral?

Yes, this method is limited to situations where the force and path of the object are circular. If the force and path are not circular, then this method may not accurately represent the work done on the object. It is important to consider the specific scenario when determining whether to use this method or not.

Similar threads

  • Introductory Physics Homework Help
2
Replies
39
Views
2K
  • Advanced Physics Homework Help
Replies
19
Views
794
  • Introductory Physics Homework Help
Replies
3
Views
2K
  • Special and General Relativity
Replies
8
Views
902
  • Precalculus Mathematics Homework Help
Replies
11
Views
2K
Replies
3
Views
1K
  • Calculus
Replies
17
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
Replies
1
Views
1K
  • Introductory Physics Homework Help
2
Replies
43
Views
2K
Back
Top