Calculating work done using line integrals

[toforfiltum]

Homework Statement

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?

Homework Equations

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!

 

[Ray Vickson]

Homework Statement

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?

Homework Equations

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!

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.

 

[toforfiltum]

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.

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