F23-Tut-08-solns

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F23 | CIVE/ENVE/GEOE/AE 115 Tutorial 8 1|9 Name: ID: Date: 1. The topographic map below shows a coal field in West Virginia. Elevations of the top of the coal deposit were observed at boreholes at points A, B and C. Assuming the top of the coal is a plane passing through the three points, basic mapping properties of the coal deposit can be determined. As shown in the figure below, the strike line is defined as the intersection of an inclined deposit and a horizontal plane. The dip angle is the angle below the horizontal plane in the direction perpendicular to the strike line (or along the water line, created in the field by pouring water down the dip). http://www.dep.wv.gov/dmr/handbooks/Pages/GeologyHandbook.aspx

F23 | CIVE/ENVE/GEOE/AE 115 Tutorial 8 2|9 a) Determine point ? along the line passing through ? and ? such that the coal elevation at ? equals the elevation at point ? . b) Determine the vector equation of the line passing through points ? and ? , which is a line with no change in elevation that lies on the top of the coal deposit. This is the strike line. c) Determine point ? on the strike line such that ?? ⃗⃗⃗⃗⃗ is perpendicular to the strike line. d) Determine the dip angle along ?? ⃗⃗⃗⃗⃗ . Report the angle in degrees below the horizontal plane. e) Determine the dip direction on the horizontal plane. Report the direction as a bearing (e.g. N20  E). a) ? = [ 750 10360 1500 ] ft ? ⃗ = [ 9050 12410 1620 ] ft ? = [ 3170 5130 1750 ] ft Direction vector for a line passing through ? and ? ?? ⃗⃗⃗⃗⃗ = ? − ? = [ 3170 5130 1750 ] − [ 750 10360 1500 ] = [ 2420 −5230 250 ] ft Equation of line passing through ? and ? 𝑃 ⃗ = ? + ??? ⃗⃗⃗⃗⃗ [ ? ? ? ] = [ 3170 5130 1750 ] + ? [ 2420 −5230 250 ] ft Find point ? such that ? ? = ? ? = 1620 ft [ ? ? ? ? ? ? ] = [ 3170 5130 1750 ] + ? [ 2420 −5230 250 ] ft ? ? = 1750 + ?(250) = 1620 ⇒ ? = 1620 − 1750 250 = −0.52 ∴ [ ? ? ? ? ? ? ] = [ 3170 5130 1750 ] + (−0.52) [ 2420 −5230 250 ] = [ 1912 7850 1620 ] ft b) Direction vector for line through points ? and ? ?? ⃗⃗⃗⃗⃗⃗ = ? ⃗ − ? ⃗ = [ 9050 12410 1620 ] − [ 1912 7850 1620 ] = [ 7138 4560 0 ] ft Vector equation of line passing through ? and ? – strike line 𝑃 ⃗ = ? ⃗ + ??? ⃗⃗⃗⃗⃗⃗ [ ? ? ? ] = [ 1912 7850 1620 ] + ? [ 7138 4560 0 ] ft

F23 | CIVE/ENVE/GEOE/AE 115 Tutorial 8 3|9 c) Find point ? on strike (line passing through ? and ? ) ?? ⃗⃗⃗⃗⃗ = ? − ? ⃗ = [ 750 10360 1500 ] − [ 1912 7850 1620 ] = [ −1162 2510 −120 ] ft ?? ⃗⃗⃗⃗⃗ || = ?? ⃗⃗⃗⃗⃗ = proj 𝐷𝐵 ⃗⃗⃗⃗⃗⃗ (?? ⃗⃗⃗⃗⃗ ) = ?? ⃗⃗⃗⃗⃗ ∙ ?? ⃗⃗⃗⃗⃗⃗ ‖?? ⃗⃗⃗⃗⃗⃗ ‖ 2 ?? ⃗⃗⃗⃗⃗⃗ = [ −1162 2510 −120 ] ∙ [ 7138 4560 0 ] ‖ [ 7138 4560 0 ] ‖ 2 [ 7138 4560 0 ] = (−1162)(7138) + (2510)(4560) + (−120)(0) 7138 2 + 4560 2 + 0 2 [ 7138 4560 0 ] = [ 314.0 200.6 0 ] ft ? ⃗ = ? ⃗ + ?? ⃗⃗⃗⃗⃗ = [ 1912 7850 1620 ] + [ 314.0 200.6 0 ] = [ 2226 8050 1620 ] ft Alternatively, ? ⃗ = ? ⃗ + ??? ⃗⃗⃗⃗⃗⃗ = [ 1912 7850 1620 ] + ? [ 7138 4560 0 ] ?? ⃗⃗⃗⃗⃗ = ? − ? ⃗ = [ 750 10360 1500 ] − ([ 1912 7850 1620 ] + ? [ 7138 4560 0 ]) = [ −1162 2510 −120 ] − ? [ 7138 4560 0 ] For ?? ⃗⃗⃗⃗⃗ to be perpendicular to ?? ⃗⃗⃗⃗⃗⃗ ?? ⃗⃗⃗⃗⃗ ∙ ?? ⃗⃗⃗⃗⃗⃗ = 0 ([ −1162 2510 −120 ] − ? [ 7138 4560 0 ]) ∙ [ 7138 4560 0 ] = 0 [ −1162 2510 −120 ] ∙ [ 7138 4560 0 ] − ? [ 7138 4560 0 ] ∙ [ 7138 4560 0 ] = 0 (3.151 × 10 6 ) − ?(71.75 × 10 6 ) = 0 ? = 0.04392 ? ⃗ = [ 1912 7850 1620 ] + (0.04392) [ 7138 4560 0 ] = [ 2226 8050 1620 ] ft

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F23 | CIVE/ENVE/GEOE/AE 115 Tutorial 8 4|9 d) ?? ⃗⃗⃗⃗⃗ = [ 2226 8050 1620 ] − [ 750 10360 1500 ] = [ −1476 2310 −120 ] ft ‖?? ⃗⃗⃗⃗⃗ ‖ = √(−1476) 2 + (2310) 2 + (−120) 2 = 2744 ft Dip angle, 𝜃 sin 𝜃 = |Δ?| ‖?? ⃗⃗⃗⃗⃗ ‖ ⇒ 𝜃 = sin −1 ( |Δ?| ‖?? ⃗⃗⃗⃗⃗ ‖ ) = sin −1 ( |−120| 2744 ) = 2.51° Alternatively, The normal vector 𝑛 ⃗ can be defined from two vectors lying on the plane passing through ? , ? , and ? . 𝑛 ⃗ = ?? ⃗⃗⃗⃗⃗⃗ × ?? ⃗⃗⃗⃗⃗ = [ 7138 4560 0 ] × [ −1476 2310 −120 ] = [ −547200 856560 23219340 ] ft 𝑛 ⃗ ∙ 𝑘 ̂ = ‖𝑛 ⃗ ‖‖𝑘 ̂ ‖ cos 𝜃 𝜃 = cos −1 ( 𝑛 ⃗ ∙ 𝑘 ̂ ‖𝑛 ⃗ ‖‖𝑘 ̂ ‖ ) = 2.51° e) ?? ⃗⃗⃗⃗⃗ ∙ 𝑗̂ = ‖?? ⃗⃗⃗⃗⃗ ‖‖𝑗̂‖ cos 𝜙 𝜙 = cos −1 ( ?? ⃗⃗⃗⃗⃗ ∙ 𝑗̂ ‖?? ⃗⃗⃗⃗⃗ ‖‖𝑗̂‖ ) = cos −1 ( [ −1476 2310 −120 ] ∙ [ 0 1 0 ] (2744)(1) ) = 32.7° ∴ Bearing of ?? ⃗⃗⃗⃗⃗ in the horizontal plane is 𝑁 32.7° 𝑊 .

F23 | CIVE/ENVE/GEOE/AE 115 Tutorial 8 5|9 2. A block weighing 500 N is pulled along a surface for a distance of 5 m at a constant velocity using an attached inclined bar as shown below. The free-body diagram shows the four force vectors involved: ? 𝑏 is the tension force in the bar, ? ? is the gravity force, ? 𝑛 is the normal force, and ? ? is the force of friction, all acting on the block. The magnitude of the tension force required to pull the block is 100 N. a) Determine the force vectors ? 𝑏 , ? ? , ? 𝑛 and ? ? . b) Show that the work done by each force over the displacement 𝑑 is given by 𝑊 = ? ∙ 𝑑 .

F23 | CIVE/ENVE/GEOE/AE 115 Tutorial 8 6|9

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F23 | CIVE/ENVE/GEOE/AE 115 Tutorial 8 7|9 3. A surface water reservoir was formed by damming a river in a valley. As shown below, the reservoir shape is approximated by a tetrahedron (3D linear shape with 4 vertices). Compute the volume of the reservoir.

F23 | CIVE/ENVE/GEOE/AE 115 Tutorial 8 8|9 Alternatively,

F23 | CIVE/ENVE/GEOE/AE 115 Tutorial 8 9|9 Alternative solution

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