www.lesswrong.com/posts/RPWoz6tYQtyinCyrn/f-ing-pulleys-how-do-they-work
2 corrections found
In this setup, you only lift 50kg of vertical weight, because you’re in a symmetrical configuration with the tree. Tada!
In a symmetric two-rope hammock, symmetry makes the two rope tensions equal, but not equal to half the weight unless the ropes are vertical. With angled ropes, each side provides only a vertical component of its tension, so the pull can be greater than 50 kg-equivalent.
Full reasoning
This mixes up equal sharing of the vertical support with equal rope tension.
In a two-rope symmetric setup, the left and right rope tensions are equal, but the ropes are angled. That means each rope contributes only its vertical component toward supporting the person’s weight. Standard static-equilibrium analysis gives
[
2T\sin\theta = W
]
when each rope makes angle (\theta) with the horizontal, so
[
T = \frac{W}{2\sin\theta}.
]
So the force you must pull with is not generally half the weight. It is half the weight only in the special case where the rope is vertical ((\theta = 90^\circ)). In a shallow hammock-like geometry, (\sin\theta) is small, so the required tension is actually greater than half the weight, and can be much greater.
That is why introductory physics examples of ropes/cables always resolve the tensions into components: the upward components add to the weight, while the horizontal components cancel. Symmetry alone does not make the pull equal to 50 kg-equivalent.
2 sources
- 5.6 Common Forces - University Physics Volume 1 | OpenStax
For the symmetric tightrope case, OpenStax derives: 2T sin 5.0° = w, so T = w / (2 sin 5.0°). It notes that the tension on either side has an upward component that supports the weight, and that the small angle results in T being much greater than w.
- Static equilibrium - Learning Lab - RMIT University
RMIT explains that in two-cable equilibrium, the upward components of the tensions add to the weight: T1 sin 30° + T2 sin 60° = W, while the horizontal components balance separately. This shows rope tension is not automatically half the weight just because there are two supports.
So yeah, you can lift the hammock a foot or two by lifting the rope over your head, and you’ll still only need ~half the lifting force you’d normally need for 100kg.
A hammock supported by two angled ropes does not generally require only half the force to lift the person. Because your pull is along the rope, the needed force depends on rope angle and is only half the weight if the rope segment is vertical.
Full reasoning
This repeats the same physics error as the earlier sentence.
When you pull on one side of a symmetric hammock, your force is the rope tension. If the rope segments are angled instead of vertical, only the vertical component of that tension lifts the person. For a symmetric two-rope support,
[
2T\sin\theta = W
]
so
[
T = \frac{W}{2\sin\theta}.
]
That means the required pull is about half the weight only in the limiting case where each supporting strand is vertical. In a typical hammock geometry, the strands are far from vertical, so the pull required is more than half the weight. In shallow hammock angles it can be dramatically more than half.
OpenStax explicitly shows this for a symmetric sagging rope: the two tensions are equal, their vertical components support the weight, and the tension rises as the rope gets more horizontal. RMIT’s statics material likewise shows that in cable systems, the components of the tensions must balance the weight, not the raw tension magnitudes themselves.
2 sources
- 5.6 Common Forces - University Physics Volume 1 | OpenStax
OpenStax states that in a symmetric sagging rope, the tension on either side has an upward component that supports the weight, and derives T = w / (2 sin θ). It also notes that as the rope becomes nearly horizontal, T becomes much greater than w.
- Static equilibrium - Learning Lab - RMIT University
RMIT explains translational equilibrium by resolving cable tensions into vertical and horizontal components; for two supporting cables, the upward components of tension add to the weight. This contradicts treating the pull in an angled rope as simply half the supported mass.