Calculating the Integral of $\vec{F}\cdot\vec{c}\,\,'$

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In summary, the conversation is discussing the calculation of a definite integral using the given vector and function. The correct answer for the integral is -1.
  • #1
jonroberts74
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Homework Statement



##\vec{c} = <\cos t, \sin t, 0>; 0 \le t \le \frac{\pi}{2}##

##\vec{F} = x\hat{i}-y\hat{j}+z\hat{k}##

Homework Equations


The Attempt at a Solution



##\vec{F}(\vec{c}(t)) = <\cos t, - \sin t, 0>##

##\vec{c}\,\,'(t) = <-\sin t, - \cos t, 0>##

##\displaystyle \int_{0}^{\pi/2} \vec{F}\cdot\vec{c}\,\,'(t)dt##

## \displaystyle \int_{0}^{\pi/2} [-\sin t\cos t + \sin t \cos t]dt = 0##

the answer should be -1 not 0 according to the book

If ## \displaystyle \int_{0}^{\pi/2} [-\sin t\cos t + \sin t \cos t]dt = 0## was ## \displaystyle \int_{0}^{\pi/2} [-\sin t\cos t - \sin t \cos t]dt = -1## then it would be correct
 
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  • #2
Check ##c'(t)##.
 
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  • #3
LCKurtz said:
Check ##c'(t)##.

ah, I know what I was doing, I was looking at ##\vec{F}(\vec{c}(t)))## when I was taking the derivative. silly
 

1. What is the purpose of calculating the integral of $\vec{F}\cdot\vec{c}\,\,'$?

The integral of $\vec{F}\cdot\vec{c}\,\,'$ is used to find the work done by a force $\vec{F}$ on an object that moves along a path $\vec{c}$.

2. How do you calculate the integral of $\vec{F}\cdot\vec{c}\,\,'$?

The integral of $\vec{F}\cdot\vec{c}\,\,'$ is calculated using the formula $\int\vec{F}\cdot\vec{c}\,\,' ds$, where $ds$ represents an infinitesimal element of the path $\vec{c}$.

3. Can the integral of $\vec{F}\cdot\vec{c}\,\,'$ be negative?

Yes, the integral of $\vec{F}\cdot\vec{c}\,\,'$ can be negative if the force $\vec{F}$ and the path $\vec{c}$ are in opposite directions, indicating that the force is doing negative work on the object.

4. What is the relationship between the integral of $\vec{F}\cdot\vec{c}\,\,'$ and the path taken?

The integral of $\vec{F}\cdot\vec{c}\,\,'$ is dependent on the path taken by the object. If the path is curved, the integral will be different compared to if the path is straight.

5. Are there any real-life applications of calculating the integral of $\vec{F}\cdot\vec{c}\,\,'$?

Yes, the integral of $\vec{F}\cdot\vec{c}\,\,'$ has practical applications in physics and engineering, such as calculating the work done by a force on a moving object, or finding the torque on a rotating body.

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