FAKULTEIT NATUURWETENSKAPPE FACULTY OF SCIENCE DEPARTMENT OF APPLIED MATHEMATICS/ DEPARTEMENT TOEGEPASTE WISKUNDE

Size: px

Start display at page:

Download "FAKULTEIT NATUURWETENSKAPPE FACULTY OF SCIENCE DEPARTMENT OF APPLIED MATHEMATICS/ DEPARTEMENT TOEGEPASTE WISKUNDE"

Alannah Thomas
7 years ago
Views:

1 FAKULTEIT NATUURWETENSKAPPE FACULTY OF SCIENCE DEPARTMENT OF APPLIED MATHEMATICS/ DEPARTEMENT TOEGEPASTE WISKUNDE MODULE CAMPUS KAMPUS APM1AEX INTRODUCTION TO STATICS INLEIDING TOT STATIKA APK APK EXAM 06 NOVEMBER 2008 EKSAMEN DATE SESSION DATUM 06/11/2008 SESSIE 09:00-12:00 ASSESSORS INTERNAL MODERATOR INTERNE MODERATOR DR M MOLELEKOA ME C SCIARONE PROF CM VILLET DURATION HOURS MARKS 100 TYDSDUUR URE PUNTE 100 NUMBER OF PAGES: 8 PAGES AANTAL BLADSYE: 8 BLADSYE INSTRUCTIONS: CALCULATORS MAY BE USED. ANSWER ALL THE QUESTIONS. SHOW ALL NECESSARY CALCULATIONS. INSTRUKSIES: SAKREKENAARS MAG GEBRUIK WORD. BEANTWOORD AL DIE VRAE. WYS ALLE NODIGE BEREKENINGE. 1

2 QUESTION 1 Let ā = ˆx 2ŷ + 2ẑ and b = 4ˆx + 5ŷ 2ẑ be two vectors VRAAG 1 Laat ā = ˆx 2ŷ + 2ẑ en b = 4ˆx + 5ŷ 2ẑ twee vektore wees. 1.1 Calculate ā b and ā b. (4) 1.1 Bereken ā b en ā b. (4) 1.2 Find the projection of b in the direction of ā. (4) 1.3 Find two unit vectors perpendicular to both ā and b. (2) 1.4 Calculate the distance of P (1, 2, 3) from the plane containing the points (0, 0, 0), (1, 2, 2) and ( 4, 5, 2). (5) 1.2 Vind die projeksie van b in die rigting van ā. (4) 1.3 Vind twee eenheidsvektore wat loodreg is op beide ā en b. (2) 1.4 Bereken die afstand tussen die punt (1, 2, 3) en die vlak wat die punte (0, 0, 0), (1, 2, 2) en ( 4, 5, 2) bevat. (5) [15] [15] QUESTION 2 VRAAG Let A( 2, 6, 7) and B( 2, 11, 5) be two points in space, relative to some XY Z- reference system. Suppose a particle is in equilibrium under the action of three forces F 1, F2 and F 3 where 2.1 Laat A( 2, 6, 7) en B( 2, 11, 5) twee punte in die ruimte wees, relatief tot n XY Z- verwysingstelsel. Veronderstel n deeltjie is in ewewig onder die invloed van drie kragte F 1, F2 en F 3 waar (a) F 1 has direction cosines respectively 0.6, 0.0 and 0.8; (b) F 2 has the same direction as the displacement AB; and (c) F 3 has x-component 42N and y-component 60N. Find the z-component of F 3. (6) [HINT: Write the forces in component form.] (a) F 1 rigtingskosinusse het van respektiewelik 0.6, 0.0 en 0.8; (b) F 2 dieselfde rigting het as die verplasing AB; en (c) F 3 n x-komponent 42N en y-komponent 60N het. Vind die z-komponent van F 3. (6) [WENK: Skryf die kragte neer in komponentvorm.] 2

3 2.2 In the figure below, a block is attached to a spring with spring constant k = 15N.m 1. The spring is unstretched in position 1. The block is pulled by a force P and remains in equilibrium in position 2. Determine (a) the magnitude of the weight of the block, (b) the magnitude of P. (10) 2.2 In die onderstaande diagram is n blok aan n veer vasgemaak met veerkonstante k = 15N.m 1. Die veer is onuitgerek in posisie 1. Die blok word dan deur n krag P verplaas na posisie 2 waar dit in ewewig verkeer. Bereken (a) die grootte van die gewig van die blok, (b) die grootte van P. (10) 25m L = 20m Block 3

4 2.3 Find the coordinates of B, C, D and E. Note that BE and AG are not parallel to the y- axis. (4) 2.3 Vind die koördinate van B, C, D en E. Neem kennis daarvan dat BE en AG nie parallel is aan die y-as nie. (4) 2m G F 2 F The 100N slider block in the diagram acted on by 2 forces F 1 and F 2 is in equilibrium but on the verge of moving to the left. The static friction coefficient of the block on the floor is 1 2. If F 1 has a magnitude of 200N, what is the magnitude of F 2? (5) [25] 2.4 Die 100N blok in die diagram is onderhewig aan 2 kragte F 1 en F 2. Die blok is in ewewig, maar op die punt om links te beweeg. Die statiese wrywingkoëffisiënt van die blok op die vloer is 1 2. As F 1 n grootte het van 200N, wat is die grootte van F 2? (5) [25] 4

5 QUESTION 3 VRAAG The following diagram shows a system of two forces and a couple. Find the resultant force and resultant couple moment around A. (10) 3.1 Die volgende diagram toon n stelsel van twee kragte en n koppel. Vind die resultante krag en resultante moment van die stelsel om A. (10) y 45N 10N _ F 1 3m _ F 2 70N A 45N 2m 6m x z 3.2 A system (not shown) can be reduced to a resultant force F = 2ˆx 3ŷ 5ẑN and resultant couple moment of ˆx 4ŷ + 2ẑNm. Show that the system can further be reduced to a single force, and find the point (0, a, b) on the line of action of this single force. (5) [15] 3.2 n Stelsel (nie getoon nie) kan gereduseer word na n resultante krag F = 2ˆx 3ŷ 5ẑN en n resultante koppel van ˆx 4ŷ + 2ẑNm. Toon aan dat die stelsel verder gereduseer kan word na n enkele krag, en vind die punt (0, a, b) op die werklyn van hierdie enkele krag. (5) [15] 5

6 Question 4 Vraag 4 B 2m 6m T 2 A C P 6m 4.1 The uniform concrete beam (weight W ) is lowered slowly with the help of two cables which are respectively fixed to the cable at A and B. Each of the cables can yield a maximum tensile force of 3W. (a) Will the cables resist the weight, and if not, which one will break first? (b) If a cable breaks, for which value of θ will this happen? [HINT: cos 2θ = 1 2 sin 2 θ] 4.1 Die uniforme betonbalk (gewig W ) word stadig laat sak met behulp van twee kabels wat respektiewelik by A en B aan die balk vasgemaak is. Die breekkrag van beide kabels is 3W. (a) Sal die kabels hou, en indien nie, watter een sal eerste breek? (b) Indien n kabel breek, vir watter waarde van θ sal dit gebeur? [WENK: cos 2θ = 1 2 sin 2 θ] 4.2 The rigid weightless L-beam in the following diagram is supported by a ball-and-socket joint at A and three cables DB, EB and F C. Calculate the tension in cable CF if the beam is loaded with a weight of 25 kn at G. Note that the AB part of the beam is along the x-axis. (13) HINT: Examine only the x-component of the resultant moment about A. 4.2 Die starre gewiglose L-balk in die volgende diagram word ondersteun deur n bal-en-potjie skarnier by A, sowel as die kabels DB, EB en F C. Bereken die spanning in kabel CF as die balk belaai word met n gewig van 2.5 kn by G. Neem kennis dat die AB gedeelte van die balk op die x-as lê. (13) WENK: Ondersoek slegs the x-komponent van die resultante moment om A. 6

7 F E C A G D B A(0,0,0) B(3,0,0) E( 0, _ 5,0 ) 4 F( C( 3,0,- 3 ) 2 _ 0, _ 5,- ) _ D( 0,0, 5 ) 4 _ G( 3,0,- _ 3 ) 4 Question Shown are two square plates, each with possible external horizontal and vertical forces acting on it. Each plate is held in place by either short rods, fixed joints or frictionless rollers. Determine for each case whether the plate is completely, over, partially or improperly constrained. (4) VRAAG Twee vierkantige plate word getoon. Elk het n moontlike eksterne belading in beide die horisontale en vertikale rigtings en word in plek gehou deur of kort stawe, vaste skarniere of wrywinglose rollers. Bepaal vir elk geval of die plaat volledig-, oor-, gedeeltelik- of onvoldoende begrens is. (4) (5.1.1) (5.1.2) _ 5.2 Determine the force in each member of the truss shown below, using the method of joints. Also indicate whether the members are in tension (T ) or compression (C). (9) 5.2 Bepaal die kragte in elk van die volgende stange van die onderstaande raamwerk deur die knooppunt metode te gebruik. Sê ook vir elke stang of dit n drukstang (DS) of trekstang (T S) is. (9) 7

8 100N 450N 4m 3m 4m 5.3 Find the force in member HI of the following truss using the method of sections. 5.3 Vind die krag in stang HI van die volgende raamwerk deur die snitmetode te gebruik. (7) [20] [20] E F 100N D C H G }Member HI 100N I B J A K o0o 8

UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS

VAN/SURNAME: UNIVERSITEIT VAN PRETORIA / UNIVERSITY OF PRETORIA DEPT WISKUNDE EN TOEGEPASTE WISKUNDE DEPT OF MATHEMATICS AND APPLIED MATHEMATICS VOORNAME/FIRST NAMES: WTW 162 DYNAMICAL PROCESSES EKSAMEN