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This is calculus based (feel free to use integrals) and we are on the concept of potential and total mechanical energy and such.

1) This new chapter we are on introduces

Work[tex]_{grav}[/tex]=mgh[tex]_{1} - [/tex]mgh[tex]_{2}[/tex] (1 being above 2) and

Work[tex]_{el}[/tex]=.5kx[tex]_{1}[/tex] [tex]^{2}[/tex] - .5kx[tex]_{2}[/tex] [tex]^{2}[/tex]

However in the previous chapter we were told that W[tex]_{net}[/tex]=[tex]\Delta[/tex]K[tex]_{energy}[/tex] or W=F[tex]\bullet[/tex] [tex]_{net}[/tex] [tex]\vec{s}[/tex]

So my question is, if the Work introduced now the same kind of work as we were told in the previous chapter?

I tried setting an quick example, lets assume we throw a ball of 10kg vertically upwards and it reaches a maximum height of 5m and than goes back down. Gravity is 10[tex]\frac{m}{s^{2}}[/tex]. So lets pretend no air resistance and such and we are trying to find work.

If I use this equation W=F[tex]_{net}[/tex][tex]\bullet[/tex][tex]\vec{s}[/tex], I get work is = to 50J

but if I use the new equation Work[tex]_{grav}[/tex]=mgh[tex]_{1} - [/tex]mgh[tex]_{2}[/tex], work is = to 500

So what is it I am not understanding? Are they 2 different kind of work?

2) Is internal energy (U[tex]_{int}[/tex]) the same as work done by non conservative forces (W[tex]_{nc}[/tex])?

edit: i have no idea why the subscripts appear as superscripts, I tried editing and fix but it still appears as superscripts. If it's not clear enough please let me know I'll make them into pictures in MSPaint and post or something