Figure 1a. A planar mechanism.
|
|
- Arabella McLaughlin
- 5 years ago
- Views:
Transcription
1 ME 5 - Machine Design I Fall Semester 0 Name f Student Lab Sectin Number EXAM. OPEN BOOK AND CLOSED NOTES. Mnday, September rd, 0 Write n ne side nly f the paper prvided fr yur slutins. Where necessary, use the figures prvided n the exam t shw vectrs instant centers. Any wrk that cannt be fllwed is assumed t be in errr. Staple each prblem separately staple yur crib sheet t the end f Prblem. Prblem (5 Pints). Part I. (5 Pints). (i) Determine the mbility f the mechanism shwn in Fig. a (number the links label the lwer higher pairs n the figure). Des the Kutzbach criterin give the crrect answer? (ii) Define suitable vectrs fr a kinematic analysis. Label shw each vectr n the figure. (iii) Write the vectr lp equatins that are required fr the kinematic analysis. Clearly identify suitable input(s) list: (a) the knwn quantities; (b) the unknwn variables; (c) any cnstraints. If yu identified cnstraints in part (c) then write the cnstraint equatins. Figure a. A planar mechanism. Part II. (0 Pints). Fr the fur-bar linkage shwn in Fig. b, the input link OA 60mm, the cupler link AB 0 mm, the utput link BO 00 mm, the grund link O O 80 mm. Determine the input angles the transmissin angles when the mechanical advantage is infinite. O Y A O Figure b. The fur-bar linkage. B X
2 ME 5 - Machine Design I Fall Semester 0 Name f Student Lab Sectin Number Prblem (5 Pints). Fr the mechanism in the psitin shwn in Fig., the distance frm the grund pivt O t pin A (cnnecting links ) is 75 mm. The length f link is OC 5 mm. The velcity f the input link is V 5 i m/s. Using the analytical methd, determine: (i) The first-rder kinematic cefficients f the mechanism. (ii) The angular velcity f link. Give the magnitude directin f this vectr. (iii) The velcity f link alng link. Give the magnitude directin f this vectr. (iv) The magnitude directin f the velcity f pint C. Shw the directin f this vectr n Fig.. Y 60 O X V = 5 m/s A C Figure. A planar mechanism.
3 ME 5 - Machine Design I Fall Semester 0 Name f Student Lab Sectin Number Prblem (5 Pints). Fr the mechanism in the psitin shwn in Fig., the angular velcity f the input link is ω 50 rad / s cunterclckwise. The mechanism is drawn full size; i.e., mm = mm. (i) List the primary instant centers list the secndary instant centers. (ii) Using the given Kennedy circle, shw the lcatin f all the instant centers n Fig.. Using the lcatin f the instant centers, determine: (iii) The first-rder kinematic cefficients f links,, 5. (iv) The angular velcities f links. Give the magnitude directin f each vectr. (v) The velcity f the slider, link 5. Give the magnitude directin f this vectr. Kennedy Circle B. C. 5 ω = 50 rad/s. O. A. O Figure. A planar mechanism. Scale: full size mm = mm.
4 ME 5 - Machine Design I Fall Semester 0 Name f Student Lab Sectin Number Prblem (5 Pints). Fr the mechanism in the psitin shwn in Fig., pin B cnnecting links is lcated vertically abve the grund pint O. The link lengths are OA 500mm, AB 00 mm, the radius f the circular grund link is 50 mm. The radius f the circular wheel which is rlling withut slipping at the pint f cntact C n the grund link is 50 mm. The cnstant angular velcity f the input link is ω 5 rad / s cunterclckwise. (i) Write a suitable vectr lp equatin fr this mechanism. Draw yur vectrs clearly n Fig.. (ii) Using yur vectr lp equatin determine the first-rder kinematic cefficients fr the mechanism. (iii) Determine the angular velcities f links. Give the magnitude directin f each vectr. (iv) Determine the velcity f pin B. Give the magnitude directin f this vectr. Y A ω = 5 rad/s ρ = 50 mm 60. C B ρ = 50 mm O O X Figure. A planar mechanism.
5 Slutin t Prblem. Part I. (i) pints. The links, the lwer pairs, the higher pairs f the mechanism are as shwn in Figure (a). j j 5 j j j j 6 j 7 j j Figure (a). The links the jints f the mechanism. The Kutzbach mbility criterin can be written as M (n ) j j () The number f links, the number f lwer pairs (r j jints), the number f higher pairs (r j jints), respectively, are n 7, j 8, j () Substituting Equatin () int Equatin (), the mbility f the mechanism can be written as Therefre, the mbility f the mechanism is M (7 ) (8) () () M () This is the crrect answer fr this mechanism, that is, fr a single input there is a unique utput. 5
6 (ii) Pints. Suitable vectrs fr a kinematic analysis f the mechanism are shwn in Figure (b). R R R R 5 R φ R R 5 R 6 R R 6 R 6 7 R Figure (b). Vectrs fr the mechanism. (iii) 7 Pints. (a) The input link is chsen here t be link (see Figure a) the input variable is the angle. Nte that link wuld be suitable as an input link since it pinned t the grund link the input variable wuld be the angle. Als, the slider (link 5) wuld be suitable as an input link the input variable wuld be the length R. 5 Anther ptin fr the input is the psitin f link 6 (the pistn) relative t link 7 (the cylinder ). Anther ptin fr the input is link 7 since it pinned t the grund link the input variable wuld be the angle. 7 Other chices fr the input link are nt really practical. (b) The six unknwn variables are the fur angles,, R 5, 6 the tw distances R R 6. (c) There are three cnstraints here, the three cnstraint equatins can be written as (a) 6 (b) (c) Since there are 6 unknwn variables then three independent vectr lp equatins are required fr a kinematic analysis f the mechanism. The three vectr lp equatins can be written as: Vectr lp is Vectr lp is I?? R RR5R 0 (5a) I C? C? R R R R R R 0 (5b) 6
7 Vectr lp is I C?? R R6 R6 R 0 (5c) Part II. 0 pints. The mechanical advantage is infinite when links are fully extended r flded n tp f each ther (tggle psitins), that is, when the angle 0 r 80. Y A β O O X Figure (c). The fur-bar linkage. B Grashf s law fr a planar fur-bar linkage, see Sectin.9, that is, Equatin (.6), page 6, states that in rder fr the fur-bar linkage t be a crank-rcker fur-bar linkage then The lengths f the fur links f the fur-bar linkage are specified as s l p q () s 60 mm (a) l 0 mm (b) p 80 mm (c) q 00 mm (d) Substituting these dimensins int Equatin () gives r 60 mm 0 mm 80 mm 00 mm (a) 80 mm 80 mm (b) Since the inequality hlds then the fur-bar linkage is a crank-rcker fur-bar linkage. Als, since the shrtest link s is adjacent t the grund link then the shrtest link is a crank, see Sectin.9. Fr the first tggle psitin, that is, when the angle 0, then Using the law f csines gives OB ABOA mm () 7
8 BO OO OB OO OBcs(80 ) (5) Substituting Equatin () BO 00 mm OO 80 mm int Equatin (5) gives Rearranging this equatin gives cs(80 ) (6a) cs(80 ) (6b) Therefre, the angular psitin f the input link is Using the law f csines, the transmissin angle can be written as (7) OO OB BO OBBO cs (8) Substituting Equatin () BO 00 mm OO 80 mm int Equatin (8) gives Rearranging this equatin gives cs (9a) cs (9b) Therefre, the transmissin angle is (0) The law f sines can als be used t determine the transmissin angle. Hwever, nte that the transmissin angle is greater than 90, that is, in the secnd quadrant. This is especially imprtant when the sine f the angle is used t determine the transmissin angle. This will give 89. which is the wrng answer. Fr the secnd tggle psitin, that is, when the angle Using the law f csines gives 8 80 then OB ABOA 80 mm () BO OO OB OO OBcs(60 ) () Substituting Equatin (7) BO 00 mm OO 80 mm int Equatin () gives cs(60 ) (a)
9 Rearranging this equatin gives cs(60 ) 8080 (a) Therefre, the angular psitin f the input link is Using the law f sines gives () 0 OB BO sin sin (60 ) (5a) Rearranging Equatin (5a) gives Substituting Equatin () OO Therefre, the transmissin angle is OB sin (60 ) sin (5b) BO 80 mm BO 00 mm int Equatin (5b) gives sin 0 (6a) 0 r 80 Fr this fur-bar linkage, the crrect answer fr the transmissin angle is (7a) 0 (7b) 9
10 Slutin t Prblem. (i) Pints. A suitable chice f vectrs fr the mechanism are as shwn in Fig. a. Y O R X V = 5 m/s C A R R Figure a. A suitable chice f vectrs fr the mechanism. The vectr lp equatin (VLE) fr the mechanism can be written as I?? R R R 0 (a) The input is the psitin f link, that is, R, the cnstraints are (b) The X Y cmpnents f the VLE, see Equatin (a), are Rcs RcsR cs 0 (a) Rsin Rsin R sin 0 (b) Differentiating Equatins () with respect t the input psitin R gives cs R cs R sin 0 (a) sin R sin R cs 0 (b) 0
11 Equatins () can be written in matrix frm as cs Rsin R cs sin Rcs sin () Using Cramer s rule, the first-rder kinematic cefficients are where the determinant is R R cs( ) DET (5a) sin ( ) DET (5b) DET R 75 mm (6) Substituting Equatins (b) (6) int Equatins (5), the first-rder kinematic cefficients are R 0.5rad/rad (7a) where the psitive sign indicates that the vectr R is getting shrter as the input link mves t the right (that is, the vectr R is getting shrter).55 rad / m (7b) where the negative sign indicates that link is rtating cunterclckwise as the input link mves t the right (that is, the vectr R is getting shrter). (ii) pints. The angular velcity f link can be written as Nte that the input velcity is R (8a) (8b) R V 5m/s that is, the vectr R is getting shrter as link mves t the right. Substituting Equatins (7b) (8b) int Equatin (8a), the angular velcity f link is The psitive sign indicates that link is rtating cunterclckwise. (iii) Pints. The velcity f link alng link can be written as (.55) ( 5) rad/ s (9) V R R (0a) Substituting Equatins (7a) (8b) int Equatin (0a), the velcity f link alng link is V ( 0.5)( 5).5 m / s (0b)
12 The negative sign indicates that link is mving upward alng link, that is, the vectr R is getting shrter fr the given input psitin velcity. (iv) Pints. The magnitude f the velcity f pint C can be written as where R the velcity vectr V C R (a) V c are shwn in Fig. b. The directin f the velcity f pint C is 0 (b) Y O X A R C V C Substituting Equatin (9) R C is Figure b. The vectr fr pint C. Alternatively. The directin f the velcity f pint C is 5mm int Equatin (a), the magnitude f the velcity f pint VC (0.5 m) 6.09 m / s (a) V c (6.09cs0 i 6.09sin 0 j).5 i 8.05 j m / s (b) Check. The X Y crdinates f pint C can be written as X Y c Rcs (a) c Rsin (b) Differentiating Equatins () with respect t the input R gives
13 X Y c Rsin (a) c Rcs (b) Substituting Equatin (8b) int Equatins () gives X c ( 0.5)(sin 0 )(.55).5 m / rad (5a) Y c ( 0.5)(cs 0 )(.55) 0.7 m / rad (5b) The velcity f pint C can be written as V c (Xci Yc j)r (6) Substituting Equatins (5) the velcity f the input link int Equatin (6), the velcity f pint C is V c (.5 i 0.7 j)( 5.00).5 i 8.00 j m / s (7a) The magnitude f the velcity f pint C is Vc m / s Nte that Equatins (7) are in gd agreement with Equatins (). (7b) Alternative vectr lp. Vectrs fr the mechanism can be chsen as shwn in Figure. Y R O X R V = 5 m/s R R A C Figure. The vectrs fr the mechanism.
14 The vectr lp equatin (VLE) fr the mechanism can be written as I?? C R R R R 0 (a) Nte that the input is the linear psitin f link, that is, R the cnstraints are 0, 80, 70 (b) (i) pints. The X Y cmpnents f the VLE, see Equatin (), are R cs R cs R cs R cs 0 (a) R sin R sin R sin R sin 0 (b) Substituting Equatins (b) int Equatins (), gives R R R cs 0 (a) R R sin 0 (b) Differentiating Equatins () with respect t the input psitin R gives R cs R sin 0 (a) R sin R cs 0 (b) Equatins () can be expressed in matrix frm as cs Rsin R sin R cs 0 (5) Using Cramer s rule, the first-rder kinematic cefficients are where R R cs DET (6a) sin DET (6b) DET R 75 mm (7) Finally, using the Equatins (6) t (7), the first-rder kinematic cefficients are R cs 0.5 rad / rad (8a)
15 sin R.55 rad / m (8b) The negative sign f R indicates that the vectr R is getting shrter as the input vectr R is getting lnger (that is, link is mving t the right) the psitive sign f indicates that link is rtating cunterclckwise as link mves t the right (that is, the vectr R is getting lnger). Nte that these answers are in agreement with the first vectr lp equatin. The angular velcity f link can be written as R (9a) Nte that R V 5m/s, that is, the vectr R is getting shrter as link mves t the right. Substituting Equatin (7b) the velcity f the input link int Equatin (8a) gives The psitive sign indicates that link is rtating cunterclckwise. The velcity f link alng link can be written as (.55) ( 5) rad/ s (9b) V R R (0a) Substituting Equatin (8a) the velcity f the input link int Equatin (0a) gives V ( 0.5)( 5).5 m / s (0b) The negative sign indicates that link is mving upward alng link, that is, the vectr R is getting shrter fr the given input psitin velcity. 5
16 Slutin t Prblem. (i) 5 Pints. The number f links is five, therefre, the ttal number f instant centers is n( n) 5 N 0 () There are 5 primary instant centers; namely, I, I, I, I 5, I 5. Therefre, there are 5 secndary instant centers; namely, I, I, I 5, I 5 I. The prcedure t lcate the secndary instant centers is: (i) The pint f intersectin f the line thrugh I I the perpendicular line t the link (where center I is lcated) is the instant center I. (ii) The pint f intersectin f the line thrugh I 5 I 5 the line thrugh I I is the instant center I. (iii) The pint f intersectin f the line thrugh I I 5 the line thrugh I I 5 is the instant center I 5. (iv) The pint f intersectin f the line thrugh I I 5 the line thrugh I I 5 is the instant center I 5. (v) The pint f intersectin f the line thrugh I I the line thrugh I I is the instant center I. I 5 at I 5 at I I 5 I I I I 5 I 5 I 5 at I I Figure. The lcatins f the instant centers. 6
17 (ii) 0 Pints. The first rder kinematic cefficient f link can be written as I I (a) II Frm Figure, the distances are measured as I I 7.80 cm I I.57 cm. Therefre, the first rder kinematic cefficient f link is I I rad/rad (b) II.57 The relative instant center I des nt lie between the tw abslute relative centers I I, therefre, the first-rder kinematic cefficient f link is psitive. The first rder kinematic cefficient f link can be written as I I (a) II Frm Figure, the distances are measured as I I 9.85 cm I I 6.89 cm. Substituting these values int Equatin (a), the first-rder kinematic cefficient f link is rad/rad (b) 6.89 The relative instant center I lies between the abslute relative centers I I, therefre, the firstrder kinematic cefficient f link is negative. The first rder kinematic cefficient f link 5 can be written as R I I (a) 5 5 Frm Figure, the distance is measured as I I 5.6 cm. Therefre, the first-rder kinematic cefficient f link 5 is R 5.6 cm/rad (b) Nte that the crrect sign f the first-rder kinematic cefficient f link 5 depends n hw the vectr R 5 is defined. If the vectr is t the left f link 5 then the first-rder kinematic cefficient f link 5 is negative if the vectr is t the right f link 5 then the first-rder kinematic cefficient f link 5 is psitive (link 5 is mving t the left as the input angle rtates cunterclckwise). (iv) 6 Pints. The angular velcity f link can be written as (5a) Substituting Equatin () the input angular velcity int Equatin (5a) gives ( 0.6)( 50.00).00 rad s (5b) The psitive sign implies that the directin f the angular velcity f link is cunterclckwise. The angular velcity f link can be written as 7
18 (6a) Substituting Equatin () the given input angular velcity int Equatin (6a) gives (.)( 50) 7.50 rad/s (6b) The negative sign implies that the directin f the angular velcity f link is clckwise (fr the given cunterclckwise angular velcity f the input link). The velcity f link 5 can be written as V R (7a) 5 5 Substituting Equatin () the given input angular velcity int Equatin (7a) gives V5 (.6)( 50).00 cm/s (7b) The negative sign implies that link 5 is mving t the left as the input link rtates cunterclckwise. (v) Pints. The velcity f pint C n link 5 is equal t the velcity f pint C n link, that is V ( I C ) ( I I ) (8a) C 5 The distance is measured as I I 5.98 cm. Substituting this measurement Equatin (6b) int Equatin (8a), the velcity f pint C is V (.98)( 7.50 rad/s).07 cm/s (8b) C The negative sign indicates that the velcity f pint C is directed t the left as the input link rtates cunterclckwise. 8
19 Slutin t Prblem. (i) 8 Pints. A set f vectrs fr a kinematic analysis f the mechanism are shwn in Fig. (a). Y A ω = 5 rad/s R R ρ = 50 mm B C 60 R 7 ρ = 50 mm O R O X Figure (a). Suitable vectrs fr the mechanism. The vectr lp equatin (VLE) can be written as I?? R RR7 R 0 () where the vectr R 7 is the arm. Nte that link des nt appear in the (VLE). Therefre, the angular velcity f link must be btained later frm the rlling cntact cnstraint. Psitin Analysis. T determine the angular psitin f link using trignmetry, see Fig. (b). 9
20 A D β B O O O Figure (b). The trignmetry f the mechanism. Using the right-angled triangle OAO, the vertical distance AO is Therefre, the distance AD is AO R sin 500sin 60.0 mm (a) AD AO DO AO BO mm (b) Using the right-angled triangle ADB, the angle ABD can be btained frm the relatin Therefre, the angle ABD is AD.0 mm sin AB 00 mm (a) (b) Therefre, the angular psitin f link is (c) (ii) 0 Pints. The X Y cmpnents f the VLE, see Equatin (), are Rcs Rcs R7cs 7 Rcs 0 (a) Rsin Rsin R7sin 7 Rsin 0 (b) Nte that the angles fr the given psitin f the mechanism are 0, 60, 7 90 (5) 0
21 Differentiating Equatins () with respect t the input psitin gives R sin R sin R sin 0 (6a) R cs R cs R cs 0 (6b) Writing Equatins (6) in matrix frm gives Rsin R7sin 7 Rsin Rcs R7cs 7 7 Rcs (7a) Substituting Equatins () (5) int Equatin (7a) gives 00sin sin sin cs cs90 500cs 60 (7b) The determinant f the cefficient matrix in Equatin (7b) is DET mm (8) Using Cramer s rule, the first-rder kinematic cefficient f link is rad / rad (9) The negative sign indicates that link is rtating clckwise fr a cunterclckwise rtatin f the input link. Similarly, the first-rder kinematic cefficient f link 7 (the arm) is rad / rad (0) The psitive sign indicates that link 7 is rtating cunterclckwise fr a cunterclckwise rtatin f the input link. (ii) 5 Pints. The angular velcity f link can be written as (a) Substituting Equatin (7) the input angular velcity int Equatin (9a) gives (.)( 5).00 rad/ s (b) The negative sign indicates that link is rtating clckwise as the input link rtates cunterclckwise. The angular velcity f link 7 (the arm) can be written as 7 7 (a) Substituting Equatin (0) the input angular velcity int Equatin (a), the angular velcity f link 7 is 7 (.7)( 5) 9.75 rad/ s (b)
22 The psitive sign indicates that the arm (link 7) is rtating cunterclckwise. The rlling cntact cnstraint between link the grund link can be written as 7 7 (a) The crrect sign is negative because there is external cntact between link link. Nting that 0 substituting Equatin (9) int Equatin (a) gives 50 (.7) 50 0 (.7) (b) Therefre, the first-rder kinematic cefficient fr link is The angular velcity f link can be written as.8 rad/rad () (5a) Substituting Equatin () int Equatin (a), the angular velcity f link is (.80 rad / rad)( 5 rad/s) 7.00 rad / s (b) The psitive sign indicates that link is rtating cunterclckwise as the input link rtates cunterclckwise. (iv) Pints. The velcity f pint B can be written as VB (a) Substituting Equatin (0b) the given input angular velcity int Equatin (a), the velcity f pint B is VB ( 9.75 rad/s)( 00 mm) 8.55 m/s (b) The psitive sign implies that pint B is mving t the left as the input link rtates cunterclckwise. Check. The psitin f pint B can be written as The X Y crdinates f pint B can be written as 7 7 I R R R () B XB Rcs Rcs (5a) YB Rsin Rsin (5b) Differentiating Equatins (5) with respect t the input gives X R sin R sin (6a) B
23 Y R cs R cs (6b) B Substituting Equatins (), (8) the input psitin int Equatins (6) gives B X ( )(sin 60 ) ( 00.00)(sin 09.0 )(.) mm / rad (7a) B Y ( )(cs 60 ) ( 00.00)(cs09.0 )(.) 0.57 mm / rad (7b) Nte here that Y B is nt zer due t the rund ff errr prpagated in the slutin. If the accurate values f the angle the first-rder kinematic cefficient are used then the first-rder kinematic cefficients fr pint B are The velcity f pint B can be written as X B 7.9 mm / rad YB V B (XBi YB j) 0mm/rad (8) Substituting Equatin (9) the velcity f the input link int Equatin (8), the velcity f pint B is V B ( i 0.57 j)( 5) i. j mm / s (9) The magnitude f the velcity f pint B is Pint B is mving t the left as the input link rtates cunterclckwise. VB 8.5m/s (0)
205MPa and a modulus of elasticity E 207 GPa. The critical load 75kN. Gravity is vertically downward and the weight of link 3 is W3
ME 5 - Machine Design I Fall Semester 06 Name f Student: Lab Sectin Number: Final Exam. Open bk clsed ntes. Friday, December 6th, 06 ur name lab sectin number must be included in the spaces prvided at
More informationEXAM 1. OPEN BOOK AND CLOSED NOTES.
ME 35 - Machine Design I Summer Semester 013 Name of Student Lab Section Number EXAM 1. OPEN BOOK AND CLOSED NOTES. Wednesday, June 6th, 013 Use the blank paper provided for your solutions. Write on one
More informationMODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b
. REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but
More informationME Machine Design I. EXAM 1. OPEN BOOK AND CLOSED NOTES. Wednesday, September 30th, 2009
ME - Machine Design I Fall Semester 009 Name Lab. Div. EXAM. OPEN BOOK AND CLOSED NOTES. Wednesday, September 0th, 009 Please use the blank paper provided for your solutions. Write on one side of the paper
More informationEXAM 1. OPEN BOOK AND CLOSED NOTES Thursday, February 18th, 2010
ME 35 - Machine Design I Spring Semester 010 Name of Student Lab. Div. Number EXAM 1. OPEN BOOK AND CLOSED NOTES Thursday, February 18th, 010 Please use the blank paper provided for your solutions. Write
More informationNamee of Student. link and I R 2 ?? = 0. and Δ θ. Calculate the. in the next. iteration. 1. = 6cm and θ * 9 = 135.
ME 52 - Machine Design I Fall Semester 2010 EXAM 1. OPEN BOOK AND CLOSED NOTES. Namee of Student Lab. Div. Number Wednesday, September 29th, 2010 Use the blank paper provided for your solutions. Write
More informationFigure 1. A planar mechanism. 1
ME 352 - Machine Design I Summer Semester 201 Name of Student Lab Section Number EXAM 1. OPEN BOOK AND CLOSED NOTES. Wednesday, July 2nd, 201 Use the blank paper provided for your solutions. Write on one
More informationPlan o o. I(t) Divide problem into sub-problems Modify schematic and coordinate system (if needed) Write general equations
STAPLE Physics 201 Name Final Exam May 14, 2013 This is a clsed bk examinatin but during the exam yu may refer t a 5 x7 nte card with wrds f wisdm yu have written n it. There is extra scratch paper available.
More informationCHAPTER 8b Static Equilibrium Units
CHAPTER 8b Static Equilibrium Units The Cnditins fr Equilibrium Slving Statics Prblems Stability and Balance Elasticity; Stress and Strain The Cnditins fr Equilibrium An bject with frces acting n it, but
More informationENGI 4430 Parametric Vector Functions Page 2-01
ENGI 4430 Parametric Vectr Functins Page -01. Parametric Vectr Functins (cntinued) Any nn-zer vectr r can be decmpsed int its magnitude r and its directin: r rrˆ, where r r 0 Tangent Vectr: dx dy dz dr
More informationYeu-Sheng Paul Shiue, Ph.D 薛宇盛 Professor and Chair Mechanical Engineering Department Christian Brothers University 650 East Parkway South Memphis, TN
Yeu-Sheng Paul Shiue, Ph.D 薛宇盛 Prfessr and Chair Mechanical Engineering Department Christian Brthers University 650 East Parkway Suth Memphis, TN 38104 Office: (901) 321-3424 Rm: N-110 Fax : (901) 321-3402
More informationDifferentiation Applications 1: Related Rates
Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm
More informationFlipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed
More informationAP Physics Kinematic Wrap Up
AP Physics Kinematic Wrap Up S what d yu need t knw abut this mtin in tw-dimensin stuff t get a gd scre n the ld AP Physics Test? First ff, here are the equatins that yu ll have t wrk with: v v at x x
More informationChapter 3 Kinematics in Two Dimensions; Vectors
Chapter 3 Kinematics in Tw Dimensins; Vectrs Vectrs and Scalars Additin f Vectrs Graphical Methds (One and Tw- Dimensin) Multiplicatin f a Vectr b a Scalar Subtractin f Vectrs Graphical Methds Adding Vectrs
More informationSubject: KINEMATICS OF MACHINES Topic: VELOCITY AND ACCELERATION Session I
Subject: KINEMTIS OF MHINES Tpic: VELOITY ND ELERTION Sessin I Intrductin Kinematics deals with study f relative mtin between the varius parts f the machines. Kinematics des nt invlve study f frces. Thus
More informationFlipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System
Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed
More informationFaculty of Engineering and Department of Physics Engineering Physics 131 Midterm Examination February 27, 2006; 7:00 pm 8:30 pm
Faculty f Engineering and Department f Physics Engineering Physics 131 Midterm Examinatin February 27, 2006; 7:00 pm 8:30 pm N ntes r textbks allwed. Frmula sheet is n the last page (may be remved). Calculatrs
More informationmaking triangle (ie same reference angle) ). This is a standard form that will allow us all to have the X= y=
Intrductin t Vectrs I 21 Intrductin t Vectrs I 22 I. Determine the hrizntal and vertical cmpnents f the resultant vectr by cunting n the grid. X= y= J. Draw a mangle with hrizntal and vertical cmpnents
More informationTrigonometric Ratios Unit 5 Tentative TEST date
1 U n i t 5 11U Date: Name: Trignmetric Ratis Unit 5 Tentative TEST date Big idea/learning Gals In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin
More informationThree charges, all with a charge of 10 C are situated as shown (each grid line is separated by 1 meter).
Three charges, all with a charge f 0 are situated as shwn (each grid line is separated by meter). ) What is the net wrk needed t assemble this charge distributin? a) +0.5 J b) +0.8 J c) 0 J d) -0.8 J e)
More information. (7.1.1) This centripetal acceleration is provided by centripetal force. It is directed towards the center of the circle and has a magnitude
Lecture #7-1 Dynamics f Rtatin, Trque, Static Equilirium We have already studied kinematics f rtatinal mtin We discussed unifrm as well as nnunifrm rtatin Hwever, when we mved n dynamics f rtatin, the
More informationM thematics. National 5 Practice Paper B. Paper 1. Duration 1 hour. Total marks 40
M thematics Natinal 5 Practice Paper B Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain
More information1 PreCalculus AP Unit G Rotational Trig (MCR) Name:
1 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Big idea In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin will invlve the unit circle which will
More information14. Which shows the direction of the centripetal force acting on a mass spun in a vertical circle?
Physics 0 Public Exam Questins Unit 1: Circular Mtin NAME: August 009---------------------------------------------------------------------------------------------------------------------- 1. Which describes
More informationEquilibrium of Stress
Equilibrium f Stress Cnsider tw perpendicular planes passing thrugh a pint p. The stress cmpnents acting n these planes are as shwn in ig. 3.4.1a. These stresses are usuall shwn tgether acting n a small
More information14. Which shows the direction of the centripetal force acting on a mass spun in a vertical circle?
Physics 3204 Public Exam Questins Unit 1: Circular Mtin NAME: August 2009---------------------------------------------------------------------------------------------------------------------- 12. Which
More informationPhys101 Final Code: 1 Term: 132 Wednesday, May 21, 2014 Page: 1
Phys101 Final Cde: 1 Term: 1 Wednesday, May 1, 014 Page: 1 Q1. A car accelerates at.0 m/s alng a straight rad. It passes tw marks that are 0 m apart at times t = 4.0 s and t = 5.0 s. Find the car s velcity
More informationPhysics 101 Math Review. Solutions
Physics 0 Math eview Slutins . The fllwing are rdinary physics prblems. Place the answer in scientific ntatin when apprpriate and simplify the units (Scientific ntatin is used when it takes less time t
More informationChapter 9 Vector Differential Calculus, Grad, Div, Curl
Chapter 9 Vectr Differential Calculus, Grad, Div, Curl 9.1 Vectrs in 2-Space and 3-Space 9.2 Inner Prduct (Dt Prduct) 9.3 Vectr Prduct (Crss Prduct, Outer Prduct) 9.4 Vectr and Scalar Functins and Fields
More informationSPH3U1 Lesson 06 Kinematics
PROJECTILE MOTION LEARNING GOALS Students will: Describe the mtin f an bject thrwn at arbitrary angles thrugh the air. Describe the hrizntal and vertical mtins f a prjectile. Slve prjectile mtin prblems.
More informationM thematics. National 5 Practice Paper C. Paper 1. Duration 1 hour. Total marks 40
N5 M thematics Natinal 5 Practice Paper C Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain
More informationFall 2013 Physics 172 Recitation 3 Momentum and Springs
Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3.
More informationSolution to HW14 Fall-2002
Slutin t HW14 Fall-2002 CJ5 10.CQ.003. REASONING AND SOLUTION Figures 10.11 and 10.14 shw the velcity and the acceleratin, respectively, the shadw a ball that underges unirm circular mtin. The shadw underges
More informationExample 1. A robot has a mass of 60 kg. How much does that robot weigh sitting on the earth at sea level? Given: m. Find: Relationships: W
Eample 1 rbt has a mass f 60 kg. Hw much des that rbt weigh sitting n the earth at sea level? Given: m Rbt = 60 kg ind: Rbt Relatinships: Slutin: Rbt =589 N = mg, g = 9.81 m/s Rbt = mrbt g = 60 9. 81 =
More informationConceptual Dynamics SDC. An Interactive Text and Workbook. Kirstie Plantenberg Richard Hill. Better Textbooks. Lower Prices.
Cnceptual Dynamics An Interactive Text and Wrkbk Kirstie Plantenberg Richard Hill SDC P U B L I C AT I O N S Better Textbks. Lwer Prices. www.sdcpublicatins.cm Pwered by TCPDF (www.tcpdf.rg) Visit the
More informationPhysics 212. Lecture 12. Today's Concept: Magnetic Force on moving charges. Physics 212 Lecture 12, Slide 1
Physics 1 Lecture 1 Tday's Cncept: Magnetic Frce n mving charges F qv Physics 1 Lecture 1, Slide 1 Music Wh is the Artist? A) The Meters ) The Neville rthers C) Trmbne Shrty D) Michael Franti E) Radiatrs
More information0606 ADDITIONAL MATHEMATICS
PAPA CAMBRIDGE CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge Internatinal General Certificate f Secndary Educatin MARK SCHEME fr the Octber/Nvember 0 series 0606 ADDITIONAL MATHEMATICS 0606/ Paper, maimum
More information1 Course Notes in Introductory Physics Jeffrey Seguritan
Intrductin & Kinematics I Intrductin Quickie Cncepts Units SI is standard system f units used t measure physical quantities. Base units that we use: meter (m) is standard unit f length kilgram (kg) is
More informationConcept Category 2. Trigonometry & The Unit Circle
Cncept Categry 2 Trignmetry & The Unit Circle Skill Checklist Use special right triangles t express values f fr the six trig functins Evaluate sine csine and tangent using the unit circle Slve tw-step
More informationChapter 2 GAUSS LAW Recommended Problems:
Chapter GAUSS LAW Recmmended Prblems: 1,4,5,6,7,9,11,13,15,18,19,1,7,9,31,35,37,39,41,43,45,47,49,51,55,57,61,6,69. LCTRIC FLUX lectric flux is a measure f the number f electric filed lines penetrating
More informationPhys101 First Major-131 Zero Version Coordinator: Dr. A. A. Naqvi Wednesday, September 25, 2013 Page: 1
Phys11 First Majr-11 Zer Versin Crdinatr: Dr. A. A. Naqvi Wednesday, September 5, 1 Page: 1 Q1. Cnsider tw unifrm slid spheres A and B made f the same material and having radii r A and r B, respectively.
More informationCHAPTER 6 WORK AND ENERGY
CHAPTER 6 WORK AND ENERGY CONCEPTUAL QUESTIONS 16. REASONING AND SOLUTION A trapeze artist, starting rm rest, swings dwnward n the bar, lets g at the bttm the swing, and alls reely t the net. An assistant,
More informationLEARNING : At the end of the lesson, students should be able to: OUTCOMES a) state trigonometric ratios of sin,cos, tan, cosec, sec and cot
Mathematics DM 05 Tpic : Trignmetric Functins LECTURE OF 5 TOPIC :.0 TRIGONOMETRIC FUNCTIONS SUBTOPIC :. Trignmetric Ratis and Identities LEARNING : At the end f the lessn, students shuld be able t: OUTCOMES
More informationCorrections for the textbook answers: Sec 6.1 #8h)covert angle to a positive by adding period #9b) # rad/sec
U n i t 6 AdvF Date: Name: Trignmetric Functins Unit 6 Tentative TEST date Big idea/learning Gals In this unit yu will study trignmetric functins frm grade, hwever everything will be dne in radian measure.
More informationPre-Calculus Individual Test 2017 February Regional
The abbreviatin NOTA means Nne f the Abve answers and shuld be chsen if chices A, B, C and D are nt crrect. N calculatr is allwed n this test. Arcfunctins (such as y = Arcsin( ) ) have traditinal restricted
More informationPhysics 321 Solutions for Final Exam
Page f 8 Physics 3 Slutins fr inal Exa ) A sall blb f clay with ass is drpped fr a height h abve a thin rd f length L and ass M which can pivt frictinlessly abut its center. The initial situatin is shwn
More informationPhysics 2010 Motion with Constant Acceleration Experiment 1
. Physics 00 Mtin with Cnstant Acceleratin Experiment In this lab, we will study the mtin f a glider as it accelerates dwnhill n a tilted air track. The glider is supprted ver the air track by a cushin
More informationKinetics of Particles. Chapter 3
Kinetics f Particles Chapter 3 1 Kinetics f Particles It is the study f the relatins existing between the frces acting n bdy, the mass f the bdy, and the mtin f the bdy. It is the study f the relatin between
More informationBuilding to Transformations on Coordinate Axis Grade 5: Geometry Graph points on the coordinate plane to solve real-world and mathematical problems.
Building t Transfrmatins n Crdinate Axis Grade 5: Gemetry Graph pints n the crdinate plane t slve real-wrld and mathematical prblems. 5.G.1. Use a pair f perpendicular number lines, called axes, t define
More informationHigher Mathematics Booklet CONTENTS
Higher Mathematics Bklet CONTENTS Frmula List Item Pages The Straight Line Hmewrk The Straight Line Hmewrk Functins Hmewrk 3 Functins Hmewrk 4 Recurrence Relatins Hmewrk 5 Differentiatin Hmewrk 6 Differentiatin
More informationPhy 213: General Physics III 6/14/2007 Chapter 28 Worksheet 1
Ph 13: General Phsics III 6/14/007 Chapter 8 Wrksheet 1 Magnetic Fields & Frce 1. A pint charge, q= 510 C and m=110-3 m kg, travels with a velcit f: v = 30 ˆ s i then enters a magnetic field: = 110 T ˆj.
More informationSections 15.1 to 15.12, 16.1 and 16.2 of the textbook (Robbins-Miller) cover the materials required for this topic.
Tpic : AC Fundamentals, Sinusidal Wavefrm, and Phasrs Sectins 5. t 5., 6. and 6. f the textbk (Rbbins-Miller) cver the materials required fr this tpic.. Wavefrms in electrical systems are current r vltage
More informationANSWER KEY FOR MATH 10 SAMPLE EXAMINATION. Instructions: If asked to label the axes please use real world (contextual) labels
ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION Instructins: If asked t label the axes please use real wrld (cntextual) labels Multiple Chice Answers: 0 questins x 1.5 = 30 Pints ttal Questin Answer Number 1
More informationInterference is when two (or more) sets of waves meet and combine to produce a new pattern.
Interference Interference is when tw (r mre) sets f waves meet and cmbine t prduce a new pattern. This pattern can vary depending n the riginal wave directin, wavelength, amplitude, etc. The tw mst extreme
More informationFinding the Earth s magnetic field
Labratry #6 Name: Phys 1402 - Dr. Cristian Bahrim Finding the Earth s magnetic field The thery accepted tday fr the rigin f the Earth s magnetic field is based n the mtin f the plasma (a miture f electrns
More informationCESAR Science Case The differential rotation of the Sun and its Chromosphere. Introduction. Material that is necessary during the laboratory
Teacher s guide CESAR Science Case The differential rtatin f the Sun and its Chrmsphere Material that is necessary during the labratry CESAR Astrnmical wrd list CESAR Bklet CESAR Frmula sheet CESAR Student
More informationQ x = cos 1 30 = 53.1 South
Crdinatr: Dr. G. Khattak Thursday, August 0, 01 Page 1 Q1. A particle mves in ne dimensin such that its psitin x(t) as a functin f time t is given by x(t) =.0 + 7 t t, where t is in secnds and x(t) is
More informationAP Physics. Summer Assignment 2012 Date. Name. F m = = + What is due the first day of school? a. T. b. = ( )( ) =
P Physics Name Summer ssignment 0 Date I. The P curriculum is extensive!! This means we have t wrk at a fast pace. This summer hmewrk will allw us t start n new Physics subject matter immediately when
More informationAQA GCSE Physics. Topic 7: Magnetism and Electromagnetism. Notes. (Content in bold is for Higher Tier only)
AQA GCSE Physics Tpic 7: Magnetism and Electrmagnetism Ntes (Cntent in bld is fr Higher Tier nly) Magnets - Nrth and Suth Ples - Same Ples repel - Oppsite ples attract Permanent Magnets - Always magnetic,
More informationSurface and Contact Stress
Surface and Cntact Stress The cncept f the frce is fundamental t mechanics and many imprtant prblems can be cast in terms f frces nly, fr example the prblems cnsidered in Chapter. Hwever, mre sphisticated
More informationNUMBERS, MATHEMATICS AND EQUATIONS
AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t
More informationPart a: Writing the nodal equations and solving for v o gives the magnitude and phase response: tan ( 0.25 )
+ - Hmewrk 0 Slutin ) In the circuit belw: a. Find the magnitude and phase respnse. b. What kind f filter is it? c. At what frequency is the respnse 0.707 if the generatr has a ltage f? d. What is the
More informationIntroduction to Smith Charts
Intrductin t Smith Charts Dr. Russell P. Jedlicka Klipsch Schl f Electrical and Cmputer Engineering New Mexic State University as Cruces, NM 88003 September 2002 EE521 ecture 3 08/22/02 Smith Chart Summary
More information1. Transformer A transformer is used to obtain the approximate output voltage of the power supply. The output of the transformer is still AC.
PHYSIS 536 Experiment 4: D Pwer Supply I. Intrductin The prcess f changing A t D is investigated in this experiment. An integrated circuit regulatr makes it easy t cnstruct a high-perfrmance vltage surce
More informationQ1. A) 48 m/s B) 17 m/s C) 22 m/s D) 66 m/s E) 53 m/s. Ans: = 84.0 Q2.
Phys10 Final-133 Zer Versin Crdinatr: A.A.Naqvi Wednesday, August 13, 014 Page: 1 Q1. A string, f length 0.75 m and fixed at bth ends, is vibrating in its fundamental mde. The maximum transverse speed
More informationMATHEMATICS Higher Grade - Paper I
Higher Mathematics - Practice Eaminatin B Please nte the frmat f this practice eaminatin is different frm the current frmat. The paper timings are different and calculatrs can be used thrughut. MATHEMATICS
More informationWYSE Academic Challenge Regional Mathematics 2007 Solution Set
WYSE Academic Challenge Reginal Mathematics 007 Slutin Set 1. Crrect answer: C. ( ) ( ) 1 + y y = ( + ) + ( y y + 1 ) = + 1 1 ( ) ( 1 + y ) = s *1/ = 1. Crrect answer: A. The determinant is ( 1 ( 1) )
More informationPhys101 Second Major-061 Zero Version Coordinator: AbdelMonem Saturday, December 09, 2006 Page: 1
Crdinatr: AbdelMnem Saturday, December 09, 006 Page: Q. A 6 kg crate falls frm rest frm a height f.0 m nt a spring scale with a spring cnstant f.74 0 3 N/m. Find the maximum distance the spring is cmpressed.
More informationNWACC Dept of Mathematics Dept Final Exam Review for Trig - Part 3 Trigonometry, 9th Edition; Lial, Hornsby, Schneider Fall 2008
NWACC Dept f Mathematics Dept Final Exam Review fr Trig - Part Trignmetry, 9th Editin; Lial, Hrnsby, Schneider Fall 008 Departmental Objectives: Departmental Final Exam Review fr Trignmetry Part : Chapters
More informationUNIT 1 COPLANAR AND NON-COPLANAR FORCES
UNIT 1 COPLANA AND NON-COPLANA FOCES Cplanar and Nn-Cplanar Frces Structure 1.1 Intrductin Objectives 1. System f Frces 1.3 Cplanar Frce 1.3.1 Law f Parallelgram f Frces 1.3. Law f Plygn f Frces 1.3.3
More informationChapter 2. Kinematics in One Dimension. Kinematics deals with the concepts that are needed to describe motion.
Chapter Kinematics in One Dimensin Kinematics deals with the cncepts that are needed t describe mtin. Dynamics deals with the effect that frces have n mtin. Tgether, kinematics and dynamics frm the branch
More information4) What is the magnitude of the net electric field at the center of the square?
Fur charges are n the fur crners f a square. Q = +5C, Q = -0C, Q 3 = +5C, Q 4 = -0C. The side length f each side f the square is 3 m. Q Q ) What is the directin f the frce n Q due t ONLY Q 4? (a) up (b)
More informationINTRODUCTION. F v. v v v v. M α M=
INTROUTION Newtn s laws and aims devised in 600 s. The cannt be prved arithmeticall. N eperimental evidence up till nw has been bserved t vilate them. These are three laws: Newtn s irst Law: bd at rest
More informationQ1. A string of length L is fixed at both ends. Which one of the following is NOT a possible wavelength for standing waves on this string?
Term: 111 Thursday, January 05, 2012 Page: 1 Q1. A string f length L is fixed at bth ends. Which ne f the fllwing is NOT a pssible wavelength fr standing waves n this string? Q2. λ n = 2L n = A) 4L B)
More informationIntroduction: A Generalized approach for computing the trajectories associated with the Newtonian N Body Problem
A Generalized apprach fr cmputing the trajectries assciated with the Newtnian N Bdy Prblem AbuBar Mehmd, Syed Umer Abbas Shah and Ghulam Shabbir Faculty f Engineering Sciences, GIK Institute f Engineering
More informationES201 - Examination 2 Winter Adams and Richards NAME BOX NUMBER
ES201 - Examinatin 2 Winter 2003-2004 Adams and Richards NAME BOX NUMBER Please Circle One : Richards (Perid 4) ES201-01 Adams (Perid 4) ES201-02 Adams (Perid 6) ES201-03 Prblem 1 ( 12 ) Prblem 2 ( 24
More informationand the Doppler frequency rate f R , can be related to the coefficients of this polynomial. The relationships are:
Algrithm fr Estimating R and R - (David Sandwell, SIO, August 4, 2006) Azimith cmpressin invlves the alignment f successive eches t be fcused n a pint target Let s be the slw time alng the satellite track
More informationCS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007
CS 477/677 Analysis f Algrithms Fall 2007 Dr. Gerge Bebis Curse Prject Due Date: 11/29/2007 Part1: Cmparisn f Srting Algrithms (70% f the prject grade) The bjective f the first part f the assignment is
More informationCHAPTER 6 -- ENERGY. Approach #2: Using the component of mg along the line of d:
Slutins--Ch. 6 (Energy) CHAPTER 6 -- ENERGY 6.) The f.b.d. shwn t the right has been prvided t identify all the frces acting n the bdy as it mves up the incline. a.) T determine the wrk dne by gravity
More informationCop yri ht 2006, Barr Mabillard.
Trignmetry II Cpyright Trignmetry II Standards 006, Test Barry ANSWERS Mabillard. 0 www.math0s.cm . If csα, where sinα > 0, and 5 cs α + β value f sin β, where tan β > 0, determine the exact 9 First determine
More informationHubble s Law PHYS 1301
1 PHYS 1301 Hubble s Law Why: The lab will verify Hubble s law fr the expansin f the universe which is ne f the imprtant cnsequences f general relativity. What: Frm measurements f the angular size and
More informationCambridge Assessment International Education Cambridge Ordinary Level. Published
Cambridge Assessment Internatinal Educatin Cambridge Ordinary Level ADDITIONAL MATHEMATICS 4037/1 Paper 1 Octber/Nvember 017 MARK SCHEME Maximum Mark: 80 Published This mark scheme is published as an aid
More information1 The limitations of Hartree Fock approximation
Chapter: Pst-Hartree Fck Methds - I The limitatins f Hartree Fck apprximatin The n electrn single determinant Hartree Fck wave functin is the variatinal best amng all pssible n electrn single determinants
More informationYou need to be able to define the following terms and answer basic questions about them:
CS440/ECE448 Sectin Q Fall 2017 Midterm Review Yu need t be able t define the fllwing terms and answer basic questins abut them: Intr t AI, agents and envirnments Pssible definitins f AI, prs and cns f
More informationPhysics 102. Second Midterm Examination. Summer Term ( ) (Fundamental constants) (Coulomb constant)
ε µ0 N mp T kg Kuwait University hysics Department hysics 0 Secnd Midterm Examinatin Summer Term (00-0) July 7, 0 Time: 6:00 7:0 M Name Student N Instructrs: Drs. bdel-karim, frusheh, Farhan, Kkaj, a,
More informationLecture 5: Equilibrium and Oscillations
Lecture 5: Equilibrium and Oscillatins Energy and Mtin Last time, we fund that fr a system with energy cnserved, v = ± E U m ( ) ( ) One result we see immediately is that there is n slutin fr velcity if
More informationChapter 3. AC Machinery Fundamentals. Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Chapter 3 AC Machinery Fundamentals 1 The Vltage Induced in a Rtating Lp e v B ind v = velcity f the cnductr B = Magnetic Flux Density vectr l = Length f the Cnductr Figure 3-1 A simple rtating lp in a
More informationQ1. In figure 1, Q = 60 µc, q = 20 µc, a = 3.0 m, and b = 4.0 m. Calculate the total electric force on q due to the other 2 charges.
Phys10 Secnd Majr-08 Zer Versin Crdinatr: Dr. I. M. Nasser Saturday, May 3, 009 Page: 1 Q1. In figure 1, Q = 60 µc, q = 0 µc, a = 3.0 m, and b = 4.0 m. Calculate the ttal electric frce n q due t the ther
More informationReview for the final exam (Math 127)
. Evaluate 3 tan tan 4 3 (b) (c) cs cs 4 7 3 sec cs 4 4 (d) cs tan 3 Review fr the final eam (Math 7). If sec, and 7 36, find cs, sin, tan, ct, csc tan (b) If, evaluate cs, sin 7 36 (c) Write the csc in
More informationMATHEMATICS Higher Grade - Paper I
Higher Mathematics - Practice Eaminatin D Please nte the frmat f this practice eaminatin is different frm the current frmat. The paper timings are different and calculatrs can be used thrughut. MATHEMATICS
More informationUGANDA ADVANCED CERTIFICATE OF EDUCATION INTERNAL MOCK 2016 PURE MATHEMATICS. 3 hours
P/ PURE MATHEMATICS PAPER JULY 0 HOURS UGANDA ADVANCED CERTIFICATE OF EDUCATION INTERNAL MOCK 0 PURE MATHEMATICS hurs INSTRUCTIONS TO CANDIDATES: Attempt ALL the EIGHT questins in sectin A and any FIVE
More informationFundamental Concepts in Structural Plasticity
Lecture Fundamental Cncepts in Structural Plasticit Prblem -: Stress ield cnditin Cnsider the plane stress ield cnditin in the principal crdinate sstem, a) Calculate the maximum difference between the
More information( ) ( ) Pre-Calculus Team Florida Regional Competition March Pre-Calculus Team Florida Regional Competition March α = for 0 < α <, and
Flrida Reginal Cmpetitin March 08 Given: sin ( ) sin π α = fr 0 < α
More informationCHAPTER 4 Dynamics: Newton s Laws of Motion /newtlaws/newtltoc.html
CHAPTER 4 Dynamics: Newtn s Laws f Mtin http://www.physicsclassrm.cm/class /newtlaws/newtltc.html Frce Newtn s First Law f Mtin Mass Newtn s Secnd Law f Mtin Newtn s Third Law f Mtin Weight the Frce f
More informationMore Tutorial at
Answer each questin in the space prvided; use back f page if extra space is needed. Answer questins s the grader can READILY understand yur wrk; nly wrk n the exam sheet will be cnsidered. Write answers,
More informationDepartment of Economics, University of California, Davis Ecn 200C Micro Theory Professor Giacomo Bonanno. Insurance Markets
Department f Ecnmics, University f alifrnia, Davis Ecn 200 Micr Thery Prfessr Giacm Bnann Insurance Markets nsider an individual wh has an initial wealth f. ith sme prbability p he faces a lss f x (0
More information= m. Suppose the speed of a wave on a string is given by v = Κ τμ
Phys101 First Majr-11 Zer Versin Sunday, Octber 07, 01 Page: 1 Q1. Find the mass f a slid cylinder f cpper with a radius f 5.00 cm and a height f 10.0 inches if the density f cpper is 8.90 g/cm 3 (1 inch
More informationMATHEMATICS SYLLABUS SECONDARY 5th YEAR
Eurpean Schls Office f the Secretary-General Pedaggical Develpment Unit Ref. : 011-01-D-8-en- Orig. : EN MATHEMATICS SYLLABUS SECONDARY 5th YEAR 6 perid/week curse APPROVED BY THE JOINT TEACHING COMMITTEE
More information