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The On Rain Forest Floor During Ecology Research

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Introduction
In an attempt to avoid walking on the rain forest floor during ecology research in South America, rope walkways were strung from tree to tree. To ensure the safety of the researchers, the maximum weight that could be carried from tree to tree without the walkways collapsing had to be set. To model this system, two pulleys containing a central object suspended halfway between them with counterweights of equal mass on the outside of the pulleys was created, as shown in Figure 1. Our goal was to determine how the sag of the walkway was related to the total mass at the center of the pulleys.

Figure 1: The apparatus used (not to scale). A and C are the masses of the counterweight objects. A and C are equal masses. B is …show more content…

F1 is the weight of counterweight A, F2 is the weight of counterweight C, and Fw is the weight of the central object. Also shown are the forces on a coordinate system for the counterweights and the central object. The distances between the pulleys and the sag of the walkway are also shown on a coordinate system.

The system is in equilibrium, so we assumed that the horizontal and vertical components of the forces will add up to zero. The counterweights A and C both had an equal mass, so they were both referred to as F1 to simplify calculations. Where Mw is the force of the mass of object B and gravity is g, we could solve for Fw. Fw = Mwg (1)

Using the mass of the counterweight objects A and C, the force F1 can be calculated using MA and gravity. Because the pulleys are frictionless the force F1 should be the same magnitude on all parts of the string between the center object and the object A.

F1 = MAg (2)

Using the trigonometry function sin, a combination of equations (1) and (2), and the law of equilibrium ΣF = 0 where Fy=0 and Fx=0, the angle ϕ can be found.

F1sinϕ + F1sinϕ = Fw (3)

Rearranging equation (3) to give the angle ϕ by itself gives ϕ = sin-1(Fw/2F1) (4)

Since the walkway system creates a right triangle, the laws of trigonometry could also be used. With the distance where the object B lies between the pulleys as L/2 and the angle ϕ, the sag of the central object, D, could be found.

D = Ltanϕ/2 (5)

Combining equations (4) and (5) to

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