I know that the work done by a particular force is defined to be:(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \int \vec{F} \cdot d\vec{r}[/tex]

and this dot product is defined as:

[tex] |F||dr| cos(\theta) [/tex]

I want to show that the work done by the force of gravity on a falling object is [tex] -\Delta U[/tex], using [tex]h_0 [/tex] and [tex] h_f [/tex] as my endpoints. So plugging in the information, I have:

[tex] \int^{h_f}_{h_0} F*cos(\theta)*dr = mg*cos(0) \int^{h_f}_{h_0} dr = mg(h_f - h_0)[/tex]

This is only the change in potential energy, rather than the negative change in potential energy. If I place a negative in front of mg to denote the direction of the force, it works out. But given the definition of the dot product where it uses absolute values, I don't see how I can do that. I'm having trouble knowing when to put in the minus sign, and when to leave it out, in other words, I'm not completely certain when it is necessary to take direction into account. If someone could explain my flawed reasoning, I would appreciate it.

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# Quick question on work

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