# Calculating work done using line integrals

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1. Nov 23, 2016

### toforfiltum

1. The problem statement, all variables and given/known data
Sisyphus is pushing a boulder up a 100-ft tall spiral staircase surrounding a cylindrical castle tower.

a) Suppose Sisyphus's path is described parametrically as $$x(t)=(5\cos3t, 5\sin3t, 10t)$$, $$\space 0\leq t\leq10$$.
If he exerts a force with constant magnitude of 50 Ib tangent to his path, find the work Sisyphus does in pushing the boulder up to the top of the tower.

b) Just as Sisyphus reaches the top of the tower, he sneezes and the boulder slides all the way to the bottom. If the boulder weighs 75 Ib, how much work is done by gravity when the boulder reaches the bottom?

2. Relevant equations

3. The attempt at a solution
OK, I'm stuck at a), so I would just type out what I have done so far.
I know that I must find $x'(t)$, so $x'(t)=(-15\sin3t, 15\cos3t, 10)$. However, this is where I'm stuck. Since usually, force is given in its $xyz$ components, it is easy for me to just do the dot product. But here, it just states that the force is 50 tangent to the path. How do I find the force in the $xyz$ components? Any hints?

Thanks!

2. Nov 23, 2016

### Ray Vickson

The total work Sisyphus performs is $W = \int \vec{F} \cdot d\vec{s}$, where $d\vec{s}$ is the arc-length along the tangent. You were told that $\vec{F}\, \| \,d\vec{s}$ at all points.

3. Nov 23, 2016

### toforfiltum

Yes, I think I've got it. $F$ is $50 \frac {x'(t)}{\left\|x'(t)\right\|}$