Work: Lifting a Load — Calc II Explorer

Units

ft · lb m · kg

Building & cable

height H = ft

cable ρ = lb/ft

The load starts on the ground at x = 0 and is winched to the roof at x = H. The cable runs from the winch down to the load, so when the load is at height x there is still H − x of cable being held up, weighing ρ(H − x). Both fields accept expressions, so 3/2 works.

Load

Constant Leaking bag Rolled poster

The load on its own — the cable's ρ(H − x) is added on top.

crate weight = lb

Constant — a crate of fixed weight. Leaking bag — a sandbag that loses r of weight for every unit it rises, so its weight is W₀ − r·x (and 0 once it has emptied), where W₀ is the bag on its own, before any cable. Rolled poster — a poster of its own length L and density λ(s), lying rolled on the ground with the cable tied to one end. Here x is the top of the poster, so the hanging length is x until the whole poster is off the ground and L thereafter, and s is measured along the poster from the tied end.

At the current height

x: —
load: —
cable: —
total F(x): —

The winch holds the load and whatever cable is still off the ground, so F(x) is the sum of the two curves beside it. Work is force times distance, so over a thin slice of the lift it is F(x)·dx — the shaded area under the total-force curve, which follows the slider.

The lift

x =

The lift

Force vs height