Physically Incorrect

This is a solution to The Climber’s Free Lunch puzzle.

The climber’s general intuition is correct. There is in fact a name for such contraptions: Bosun’s chairs.  Let’s start by drawing a force diagram for each case:

The equations of motion for the two cases are

• Left: MA = -Mg + T
• Right: MA = -Mg + 2T

where M is the climber’s mass and T is the tension in the rope (remember the rope is massless and hence “transfers” the tension throughout its length). For a constant ascension rate, A=0 (no acceleration), and we obtain:

• Left: T = Mg
• Right: T = Mg/2

This tells us something quite interesting, conceptually: if we desire to move up at a constant rate, we’re going to have to apply a different amount of force in each case. What if the climber were to apply the same amount of force in the second case (right) as in the first (left)? Well, this would result in T = Mg, ergo the climber would accelerate upwards: MA = -Mg + 2T = -Mg + 2Mg = Mg –> A = g.Therefore, beware of the following sort of reasoning: “in the first case the climber will apply a force F which will generate a tension T in the rope; in the second he will apply the same force and move up twice as fast.” This implicitly - and wrongly - assumes the same amount of force is applied in both scenarios. In fact, the equations of motion tell us that, in general, having the climber apply a constant force F (by using his muscles) will result in accelerations given by:

• Left: A = -g + F/M
• Right: A = -g + 2F/M

The acceleration in the second case is not, in general, twice that in the first case, and doubling the amount of force (in each case) will not double the acceleration!

Will there be a price to pay in the second case? Well, yes. The rope is twice as long in the second case; to climb to the top, the climber will need to pull on twice the length. As the rate at which an average climber can climb up a rope is more or less fixed, the velocity will be halved, so it will take him twice the amount of time to reach the top. That’s the tradeoff with pulleys: they make things easier, but more time consuming.

Finally, beware of a general conceptual mistake made by many students which confuse mechanical force with the subjective feeling of ‘force’. Even in equilibrium the climber still needs to hang on to the rope which requires quite a bit of effort. On the other hand, simply standing on the ground requires almost no effort at all, and yet from a Newtonian mechanics point of view, both equilibria will have the same force diagrams (in the first, it will be the tension in the rope that cancels out with gravity, and in the second it will be the normal force created by the floor).