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A point mass attached to a string rotates. The string goes through a smooth tube and is pulled slowly, thus encreasing the velocity of the mass. See drawing. The solution is taken from a book.

The work done by the string is:

[tex]W=\int_{r_0}^{r}\frac{L^2}{m}\frac{1}{r^3}\left(-dr\right)=+\frac{L^2}{m}\int_{r}^{r_0}\frac{1}{r^3}dr=\frac{L^2}{2m}\left(\frac{1}{r^2}-\frac{1}{r_0^2}\right)[/tex]

I ask about the minus and plus signs in the first integral (i understand the physics).

Let's say the origin of the axes system is in the center, pointing outward. Then, I understand, the minus sign of the (-dr) is because it is directed to the negative direction, to the center, since the radius decreases. But if so, why is the force:

[tex]F=\frac{L^2}{m}\frac{1}{r^3}[/tex]

taken as positive? it should have been also negative, since work equals to:

[tex]\vec{W}=\vec{F}\cdot\vec{S}[/tex]

And amazingly the result is correct: the work is positive.

I think the reasoning here is mathematical, not physical, since later on the borders of the integral [itex]\int_{r_0}^{r}[/itex] switch to [itex]\int_{r}^{r_0}[/itex], and together with the changing of the sign of the dr, both give a meaningful expression, although in the opposite direction: from r to r_{0}.

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# Homework Help: Signs in the integral of work

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