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**
1. Homework Statement **

The tension in the cable is 800LBs. Find the depth of water that produces this tension. The gate is hinged at B; the cable is attached at A. (Figure 1)

The tension in the cable is 800LBs. Find the depth of water that produces this tension. The gate is hinged at B; the cable is attached at A. (Figure 1)

## Homework Equations

## The Attempt at a Solution

OK, so this is just 2 opposing lever arms: the tension in the cable is pulling the gate closed (clockwise) and the pressure of the water is trying to open the gate (counter-clockwise).

So we start on the easy side.

Given the geometry, we know that the gate forms a 3/4/5 triangle and therefore the angle between the gate and the vertical is 36.8 degrees (Figure 5).

An 800-LB pull on the cable translates into a 480-LB pull perpendicular to the gate (Figure 2).

The cable's lever arm is therefore (480)(2.5)=1200 Ft-LBs.

This is opposed by the force of the water.

OK, so I cheated and looked at the answer in the book, and I still cannot see how they got this answer.

The book says that the water is 5.18 feet deep; we call this h.

h/3 is 1.727 feet; this is the centroid of the body of water.

So 1.727 feet from the bottom takes us .727 feet into our 3/4/5 triangle (Figure 4), which means that at this point the lever arm for the water is 1.6 (Figure 3).

The force of the water is (1/2)(gamma x h)(h) which is (1/2) (62.4 x 5.18)(5.18).

This equals 837 Lbs/Ft squared.

The torque exerted by the water is therefore (837)(1.6) = 1339 Ft-Lbs

And we were expecting 1200.

Where did I make a wrong turn?

Thank you in advance for any help

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