Eagle's Nest

Basic Thermodynamics ~ J. Pohl ©

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To observe eagles nesting, naturalists built a blind. With a cross-bow, a line was shot through the canopy of a neighboring tree. Next, a block and tackle ("gun tackle" ~ shown below) was hoisted, attached to attach to a secure limb. With this equipment they hoisted themselves to the blind.

First Lift: The woman dons the harness and bedecks herself with the equipment. What must be done is seen by the configuration of the pulley system (Figure @). They have done this before; the man hauls rope to lift her to the tree top (m_womang_o = 500 N). Use the information of sketches.

i) Calculate the Work: Use integration of the hauling force applied by the man, through the rope displacement as it passes through his hands.

The scenario (right) and arrangement of the gun tackle (below right) are shown. For this hoist we take the system to be the tackle, woman and the equipment she carries. A system schematic is shown below right.

♦ Work by Integration: The general arrangement of the gun tackle with the woman being hoisted is shown ( left).

Needing the force required of the man, we select a system to be the rope, tackle and woman . A free body sketch is drawn, the purpose of which is to determine the pulling force the man must effect during the lift. The momentum equation is applied is:

(1) 1

Since there are three forces, the force of the man cannot be determined. To prove this fact, realizing also that the man might pull at an angle from the vertical, we write the forces in component form and multiply by the unit vectors, I the K.

(2)2

The above set of equations are not solvable. We are obliged to try again with a "smaller" system as drawn below right.

In the smaller free-body-diagram we have cut through the ropes of the pulley: The rope tensions of either side of the pulley are equal because the pulley is assumed frictionless. A sum of moments about the pulley center will show the tensions equal. Thus we have:

0 = T_L + T _R - 500N

and T_L = T_R so T = 250N

(3) 3

Consequently, to effect the lift, the man must apply a force of 250 N along the line of the rope. The next task is to determine the length of rope he must pull. The sketch (below right) showing "before" and "after" of the hoist and makes it clear that the man must pull 60 meters of rope with a force of 250 Newtons.

Now, knowing F_pull, we return to the original system (the rope, tackle and woman). The man, (part of the surroundings) applies force to the boundary of the system (at successive places on the surface of the rope). With effort he displace parts of the system boundary downward, grabs a next section of boundary, and so on. For the pulling event we define the direction the rope moves to have a unit vector parallel to the rope. Imagine the vector to be just above the man's grip, parallel to the rope, pointing toward his hands. Call that unit vector, e_rope. With this unit vector, we can write the force applied to the boundary of the rope and the displacement of the rope as needed by the work integral and then integrate it.

(4) 4

♦ Work by Energy Equation: We select the system to the woman, tackle and ropes. The energy change of the system equals the work supplied by the man.

(5) 5

Complete System With the man, the woman and the ropes taken as the system, what conclusion does the energy equation provide?
♦ We use the extrinsic energy equation combined with the intrinsic equation.

(6) 6

We went ahead and put friction in this equation. Maybe the pulley gets leaves caught in it, or the apparatus snags on a branch, or his grip slips - a lot could go wrong. The man exerts arduous metabolic effort but he does no thermodynamic work. He experiences change of internal energy. His ΔU is negative and at the very least equal in magnitude to the increase of potential energy (ΔPE) of the woman. Also the man gets hot and perspires.

There is heat - from the system, ΣQ = Q_1-2 is negative. The above is a realistic representation. ΔPE of the woman is a positive number - that energy must be paid by physical effort of the man.

Second Lift: The sketch shows the man in the process of hoisting himself the the blind. For the man, mg_o equals 900 Newtons. Calculate the force required of the man and the length of rope he must haul.

In a manner similar to the hoist of the woman, we see the force must be 300 Newtons and that 90 meters of rope must be hauled.

With the man and ropes as system, what does the energy equation tell us?

Sorry! This is incomplete!