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Chapter 6 | Applications of Integration
Figure 6.57 A thin plate submerged vertically in water.
Let’s now estimate the force on a representative strip. If the strip is thin enough, we can treat it as if it is at a constant depth, s ( x i * ). We then have F i = ρAs = ρ ⎡ ⎣ w ( x i * )Δ x ⎤ ⎦ s ( x i * ). Adding the forces, we get an estimate for the force on the plate: F ≈ ∑ i =1 n F i = ∑ i =1 n ρ ⎡ ⎣ w ( x i * )Δ x ⎤ ⎦ s ( x i * ). This is a Riemann sum, so taking the limit gives us the exact force. We obtain (6.13) F = lim n →∞ ∑ i =1 n ρ ⎡ ⎣ w ( x i * )Δ x ⎤ ⎦ s ( x i * ) = ∫ a b ρw ( x ) s ( x ) dx . Evaluating this integral gives us the force on the plate. We summarize this in the following problem-solving strategy. Problem-Solving Strategy: Finding Hydrostatic Force 1. Sketch a picture and select an appropriate frame of reference. (Note that if we select a frame of reference other than the one used earlier, we may have to adjust Equation 6.13 accordingly.) 2. Determine the depth and width functions, s ( x ) and w ( x ). 3. Determine the weight-density of whatever liquid with which you are working. The weight-density of water is 62.4 lb/ft 3 , or 9800 N/m 3 . 4. Use the equation to calculate the total force.
Example 6.27 Finding Hydrostatic Force
A water trough 15 ft long has ends shaped like inverted isosceles triangles, with base 8 ft and height 3 ft. Find the force on one end of the trough if the trough is full of water.
Solution
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