Line integral to calculate work

In summary, the conversation discusses a problem that requires calculating the work done by force P and converting the variable x into theta. The solution involves differentiating both sides with respect to theta and using the concept of energy and conservative forces.
  • #1
Hernaner28
263
0
Hi. I have a concrete doubt with this problem. Here's the pic.

attachment.php?attachmentid=46109&stc=1&d=1334149747.jpg


It asks me to calculate the work done by force P (the ball moves with constant speed). So the solution is in the book and I understood everything, but the problem comes here,

the force P in axis y is zero so the work of P should be:

[tex]\int\limits_{{x_o}}^{{x_f}} {{P_x}dx} [/tex]

And we know that Px is equal to the tension in axis x so:

[tex]\int\limits_{}^{} {T\sin \theta dx} [/tex]

But we need to convert the variable dx into theta variable. And the books states that as:
[tex]x = L\sin \theta [/tex]
then:
[tex]dx = L\cos \theta d\theta [/tex]

But shouldn't it be:
[tex]dx = L\sin\theta d\theta [/tex]??

Why did it take the derivative of sine and not of x on the other side? Thank you!
 

Attachments

  • resnick.jpg
    resnick.jpg
    21.5 KB · Views: 548
Physics news on Phys.org
  • #2
Given:
##~~~x = L sin(\theta)##

Differentiate w.r.t. θ:
##~~~\frac{dx}{d \theta} = L cos(\theta) ##

Rearrange:
##~~~dx = L cos(\theta) d\theta##
 
  • #3
Sorry but what did you do in the second step (wrt?)? I think I'm understaning now but I still don't get it.

Thanks!
 
  • #4
Differentiate both sides with respect to theta.
 
  • #5
Whovian said:
Differentiate both sides with respect to theta.

Aham! So it would be an intermediate step there:

[tex]\frac{{dx}}{{d\theta }} = \frac{d}{{d\theta }}L\sin \theta [/tex]

I didn't know that you could take dO to the other side as a product. Thanks!
 
  • #6
Think about it. If two functions are constantly the same, shouldn't their derivatives also be equal?
 
  • #7
And could this exercise be solved easily using the concepts of energy and conservative forces? Then, how could I begin?

Thanks!

Edit. I'll create a new thread.
 

1. What is a line integral?

A line integral is a type of integral used to calculate the work done by a force along a specific path. It takes into account both the magnitude and direction of the force, as well as the displacement along the path.

2. How is a line integral used to calculate work?

A line integral is used to calculate work by integrating the dot product of the force and displacement vectors along a specific path. This takes into account the work done by the force in the direction of the displacement.

3. What are the applications of using a line integral to calculate work?

Line integrals are commonly used in physics and engineering to calculate the work done by a force, such as in the study of fluid flow or electromagnetism. They can also be applied in other areas such as economics and finance to calculate the work done by a changing quantity.

4. How is a line integral different from a regular integral?

A line integral is different from a regular integral because it takes into account the path along which the integration is done. A regular integral integrates over a single variable, while a line integral integrates over a curve or path.

5. What are the different types of line integrals?

There are two main types of line integrals: path integrals and contour integrals. Path integrals are used to integrate along a specific path, while contour integrals are used to integrate over a closed curve in the complex plane. These types of integrals are commonly used in physics, engineering, and mathematics.

Similar threads

  • Introductory Physics Homework Help
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
10
Views
800
  • Introductory Physics Homework Help
Replies
4
Views
751
  • Introductory Physics Homework Help
Replies
17
Views
279
  • Introductory Physics Homework Help
Replies
9
Views
829
  • Introductory Physics Homework Help
Replies
1
Views
199
  • Introductory Physics Homework Help
Replies
1
Views
117
  • Introductory Physics Homework Help
Replies
13
Views
2K
  • Calculus and Beyond Homework Help
Replies
5
Views
596
Replies
7
Views
217
Back
Top