1 Lb 6 ROTATIONS (163 pts totl) Eercise 1: Apprent dip problem (24 points totl) 1) An pprent dip of 62 to the southwest is mesured for bedding plne in verticl cross section tht strikes 230 (cll this pprent dip vector 1). An pprent dip of 34 to the southest is mesured for bedding plne in verticl cross section tht strikes 110 (cll this pprent dip vector 2). Wht is the strike of the bedding plne nd the true dip of the bedding plne? To solve this problem ou find the common plne tht contins two intersecting lines (which in this cse re the two vectors showing pprent dips). Solve the problem using n equl-ngle sphericl projection (7 points totl; 2 points for plotting ech line, 1 pt for grphicll identifing the common plne, nd 2 pts for getting the strike nd dip of the common plne) Strike Dip 1b) Solve the problem with cross products using Mtlb. Include cop of our Mtlb printout. (17 points totl, 1 point per bo) Trend Plunge α β γ Vector v1 Vector v2 N=v2v1 Components of cross product v2 v1 N N N Pole N Pole N Plne Plne trend ( ) plunge ( ) strike ( ) dip ( ) Stephen Mrtel Lb 6-1 Universit of Hwii
2 Eercise 2: Rottion problem 2 (15 points totl) 2) An outcrop displs regulr set of current ripple mrks. The es of the ripple mrks pitch 32 northwest in the bedding plne, nd the bedding plne strikes 325 nd dips 20 NE. Determine the direction of the originl pleocurrents responsible for the ripple mrks b restoring the beds bck to horiontl (ssume the current flowed perpendiculr to the es of the ripples. Before ou nswer tht question, first determine the orienttion of the rottion is N nd the ngle of rottion θ. Netl lbel our stereonets to show how the relevnt fetures rotte (i.e., the ripple is nd the pole to bedding). (1 pt/bo here; 5 subjective points for clrit of stereonet work) Trend of Plunge of Rottion ngle ω ( ) rottion is θ ( ) rottion is φ ( ) Eisting ripple is trend ( ) Eisting ripple is plunge ( ) Restored ripple is trend ( ) Restored ripple is plunge ( ) Originl trend of current ( ) Originl plunge of current ( ) Rke of restored ripple is ( ) Aes of ripple crests (mp view) Oscillting current direction Stephen Mrtel Lb 6-2 Universit of Hwii
Eercise 3: Rottions using liner lgebr A common problem in geolog involves trnsforming dt from one reference frme to nother. A simple but informtive emple involves chnging coordintes from n = north, = est, = down reference frme to n = est, = north, = up reference frme. This rottion problem usull is highl inconvenient to do using sphericl projections but cn be done rpidl using three mtri methods discussed in clss. 3 1 Drw n oblique perspective 3D digrm showing ech of the reference frme, lbeling the es of the two reference frmes nd the directions north, est, up, nd down. 10 pts totl; 5 pts for ech reference frme, with 3 pts for es nd 2 pts for lbels) 2 Rottion using direction cosine mtrices. Complete the following tbles, entering the ngles in degrees, between the es in the unprimed nd triple- primed reference frmes. The ngle θ is the ngle from the - is to the - is, nd is the cosine of the ngle from the - is to the - is. (36 pts totl; 1 pt/bo) θ = θ = θ = θ = θ = θ = θ = θ = θ = θ = θ = θ = θ = θ = θ = θ = θ = θ = Rottion mtri for converting triple- primed coordintes to unprimed coordintes = = = = = = = = = Rottion mtri for converting unprimed coordintes to triple- primed coordintes = = = = = = = = = Stephen Mrtel Lb 6-3 Universit of Hwii
3 Construction of rottion mtri using consecutive rottions. (15+54+9 = 78 pts) 4 A On seprte pge, drw sequence of three drwings showing the three sequentil rottions bout the -, -, nd - es needed to rotte the,, es to the,, es. The sequence should illustrte the rottions described below, nd it should stte the ngle of rottion. The drwings should be net nd clerl lbeled (15 points totl, 5 points/drwing). First rottion: bout -is to bring trend of - is into lignment with trend of - is. θ = θ = θ = θ = θ = θ = θ = θ = θ = First rottion mtri R1; converts unprimed () coordintes to single- primed ( ) coordintes = = = = = = = = = Second rottion: bout -is to bring - is into lignment with - is. θ = θ = θ = θ = θ = θ = θ = θ = θ = Second rottion mtri R2; converts ( ) coordintes to ( ) coordintes = = = = = = = = = Third rottion: bout -is to bring ll ( ) es into lignment with ll ( ) es. θ = θ = θ = θ = θ = θ = θ = θ = θ = Third rottion mtri R3; converts ( ) coordintes to ( ) coordintes = = = = = = = = = Single mtri for doing rottion using one rottion: R = R3*R2*R1 Include cop of our Mtlb work to do this net step. R = R3 * R2 * R1 = * * Stephen Mrtel Lb 6-4 Universit of Hwii
4 Construction of rottion mtri using consecutive rottions. (30 pts) In order to bring the unprimed reference frme into coincidence with the triple- primed reference frme in one step, wht re the trend nd plunge of the rottion is, nd wht is the ngle of rottion? You should be ble to visulie this. Mke sure ou get the sign of the rottion ngle correct using right- hnd rule for positive ngle. 5 Trend ( ) Plunge ( ) Rottion ngle ( ) To four significnt figures, wht re the direction cosines (,, ) of the rottion is? Using the formuls on pge 11 of lecture 10, (1- cosθ) + cosθ (1- cosθ) + (1- cosθ) - sinθ (1- cosθ) - (1- cosθ) + cosθ (1- cosθ) + sinθ (1- cosθ) + (1- cosθ) - (1- cosθ) + cosθ Clculte the single- step rottion mtri R for converting unprimed coordintes to triple- primed coordintes. Show our work on seprte pge; Mtlb printout is preferred. = = = = = = = = = 4 Conversion of coordintes. Consider the,, coordintes listed column- b- column below in mtri X (ech of the 5 points hs,, coordintes) X Point 1 Point 2 Point 3 Point 4 Point 5 1 2-1 1 0 1 2-1 2 0 1 2-1 3 0 Use the one- step rottion mtri to convert the,, coordintes to,, coordintes b pre- multipling them en msse using the rottion mtri nd Mtlb (i.e., X = RX). Show our Mtlb work X Point 1 Point 2 Point 3 Point 4 Point 5 ' Stephen Mrtel Lb 6-5 Universit of Hwii