When Sisyphus sneezed ...

Question

Sisyphus is pushing a boulder up a 90 ft tall spiral staircase surrounding a cylindrical castle tower

of radius 20 ft. Sisyphus makes exactly three revolutions to reach the top while he exerts a force

with a constant magnitude of 50 lb on the boulder tangent to his path.

A. 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 75lb how much work is done by gravity when the boulder

reaches the bottom

Answer 1

Depending on whether Sisyphus is right handed or left handed, the path

is p(t)=[x(t)=20cos(2πt), y(t)=±20sin(2πt), z=30t], but since the force vector

is given as tangent, the inner product of F vector and ds is (since cosθ=1)

equal to |F|*|ds| (no need to add vector terms)

integrating from t= 0 to 3 (90 feet, 3 rotations)

work is 3*50*√(40²π²+30²) pound-feet

where

ds=√[(-40πcos(2πt))²+(±-40πcos(2πt))²+30²)]

on way down the force vector is known as (0,0,75)

the dot product with ds is 75*30

so the work done by gravity on way down is integral 75*30dt ,,,t=0 t0 t=3

or 75*90