# Chapter 6. Gradually-Varied Flow in Open Channels

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapte 6 Gadually-Vaied Flow in Open Channels 6.. Intoduction A stea non-unifom flow in a pismatic channel with gadual changes in its watesuface elevation is named as gadually-vaied flow (GVF). The backwate poduced by a dam o wei acoss a ive and dawdown poduced at a sudden dop in a channel ae few typical examples of GVF. In a GVF, the velocity vaies along the channel and consequently the bed slope, wate suface slope, and enegy line slope will all diffe fom each othe. Regions of high cuvatue ae excluded in the analysis of this flow. The two basic assumptions involved in the analysis of GVF ae:. The pessue distibution at any section is assumed to be hydostatic. This follows fom the definition of the flow to have a gadually vaied wate suface. As gadual changes in the suface cuvatue give ise to negligible nomal acceleations, the depatue fom the hydostatic pessue distibution is negligible.. The esistance to flow at any depth is assumed to be given by the coesponding unifom flow equation, such as the Manning equation, with the condition that the slope tem to be used in the equation is the enegy line slope, S e and not the bed slope, S. Thus, if in a GVF the depth of flow at any section is y, the enegy line slope S e is given by, n V S e (6.) 4 3 R whee R hydaulic adius of the section at depth y. 6.. Basic Diffeential Equation fo the Gadually-Vaied Flow Wate Suface Pofile Since, S Channel bed slope fo unifom flow depth y, Wate depth vaiation fo the canal each, d (V /g) Velocity head vaiation fo the each. Witing the enegy equation between the coss-sections and (Fig. 6..),

2 V S + y + g + + ( y + ) V S e + d g S S S e e e d V g d V g V V + + d + Se g g S e (6.) d V + g y V E.L S e d V /g+d(v /g S d y+d d V+d S Figue 6.. V g Q ga ( y) da T ( y) d Q ga V A g T Substituting this to Equ. (6.), ( y) ga( y) ( y) ( y) Q A F e F da 4 Q T ga S (6.3) ( y) 3 ( y)

3 Equ. (6.3) is the geneal diffeential equation of the wate suface pofile fo the gadually vaied flows. / gives the vaiation of wate depth along the channel in the flow diection Classification of Flow Suface Pofiles Fo a given channel with a known Q Dischage, n Manning coefficient, and S Channel bed slope, y c citical wate depth and y Unifom flow depth can be computed. Thee ae thee possible elations between y and y c as ) y > y c, ) y < y c, 3) y y c. Fo hoizontal (S ), and advese slope ( S < ) channels, Q A R n 3 S Hoizontal channel, S Q, Advese channel, S <, Q cannot be computed, Fo hoizontal and advese slope channels, unifom flow depth y does not exist. Based on the infomation given above, the channels ae classified into five categoies as indicated in Table (6.). Table 6.. Classification of channels Numbe Channel categoy Symbol Chaacteistic condition Remak Mild slope M y > y c Subcitical flow at nomal depth Steep slope S y c > y Supecitical flow at nomal depth 3 Citical slope C y c y Citical flow at nomal depth 4 Hoizontal H S Cannot sustain unifom flow bed 5 Advese slope A S < Cannot sustain unifom flow Fo each of the five categoies of channels, lines epesenting the citical depth (y c ) and nomal depth (y ) (if it exists) can be dawn in the longitudinal section. These would divide the whole flow space into thee egions as: Region : Space above the topmost line, Region : Space between top line and the next lowe line, Region 3: Space between the second line and the bed. 3

4 Figue (6.) shows these egions in the vaious categoies of channels. Figue 6.. Regions of flow pofiles Depending upon the channel categoy and egion of flow, the wate suface pofiles will have chaacteistics shapes. Whethe a given GVF pofile will have an inceasing o deceasing wate depth in the diection of flow will depend upon the tem / in Equ. (6.3) being positive o negative. S (6.3) e F 4

5 Fo a given Q, n, and S at a channel, y Unifom flow depth, y c Citical flow depth, y Non-unifom flow depth. The depth y is measued vetically fom the channel bottom, the slope of the wate suface / is elative to this channel bottom. Fig. (6.3) is basic to the pediction of suface pofiles fom analysis of Equ. (6.3). Figue 6.3 To assist in the detemination of flow pofiles in vaious egions, the behavio of / at cetain key depths is noted by stuing Equ. (6.3) as follows: y > y y < y y y c c c F F F < > And also, y > y y < y y y S S S e e e < S > S S. As y y, V V, S e S S lim y y F cons The wate suface appoaches the nomal depth asymptotically. 5

6 . As y yc, F, F, S e S lim y y c F e The wate suface meets the citical depth line vetically. 3. As y, V F S e lim y S F e S S The wate suface meets a vey lage depth as a hoizontal asymptote. Based on this infomation, the vaious possible gadually vaied flow pofiles ae gouped into twelve types (Table 6.). Table 6.. Gadually Vaied Flow pofiles Channel Region Condition Type Mild slope 3 Steep slope 3 Citical slope 3 Hoizontal bed 3 Advese slope 3 y > y > y c y > y > y c y > y c > y y> y c > y y c > y > y y c > y > y y > y y y < y y c y > y c y < y c y > y c y < y c M M M 3 S S S 3 C C 3 H H 3 A A 3 6

7 6.4. Wate Suface Pofiles M Cuves Figue 6.4 Geneal shapes of M cuves ae given in Fig. (6.4). Asymptotic behavios of each cuve will be examined mathematically. a) M Cuve Wate suface will be in Region fo a mild slope channel and the flow is obviously subcitical. S e < S Mild slope channel y > y c Subcitical flow S e F F < Subcitical flow ( F ) > y > y S e < S + + > (Wate depth will incease in the flow diection) Asymptotic behavio of the wate suface is; Wate depth can be between ( > y > y ) fo Region. The asymptotic behavios of the wate suface fo the limit values (, y ) ae; 7

8 a) y, V, F, ( F ) y, V, S e lim y S S The wate suface meets a vey lage depth as a hoizontal asymptote. b) y y, V V, S e S S lim y y F The wate suface appoaches the nomal depth asymptotically. The most common of all GVF pofiles is the M type, which is a subcitical flow condition. Obstuctions to flow, such as weis, dams, contol stuctues and natual featues, such as bends, poduce M backwate cuves (Fig. 6.5). These extend to seveal kilometes upsteam befoe meging with the nomal depth. b) M Cuve Figue 6.5. M Pofile Wate suface will be in Region fo a mild slope channel and the flow is obviously subcitical. (Fig. 6.4). y > y > y c y < y V > V S e > S (S S e ) < y > y c F < ( F ) > S F e + (Wate depth decease in the flow diection) 8

9 Asymptotic behavio of the wate suface is; y > y > y c y y, S e S, (S S e ) S lim y y F The wate suface appoaches the nomal depth asymptotically. y y c, F, ( F ) S lim y yc e The wate suface meets the citical depth line vetically. The M pofiles occu at a sudden dop of the channel, at constiction type of tansitions and at the canal outlet into pools (Fig. 6.6). c) M 3 Cuve Figue 6.6 Wate suface will be in Region 3 fo a mild slope channel and the flow is obviously subcitical. (Fig. 6.4). y > y c > y < y V > V S e > S (S S e ) < y < y c Supecitical flow F > ( F ) < S F e + (Wate depth will incease in the flow diection) Asymptotic behavio of the wate suface is; 9

10 y y c F ( F ) S e lim y yc The wate suface meets the citical depth line vetically. y, S e, (S S e ) y, F V gy lim (Unknown) y y The angle of the wate suface with the channel bed may be taken as S. y c 3 Figue 6.7. M 3 Pofile Whee a supecitical steam entes a mild slope channel, M 3 type of pofile occus. The flow leading fom a spillway o a sluice gate to a mild slope foms a typical example (Fig. 6.7). The beginning of the M 3 cuve is usually followed by a small stetch of apidly vaied flow and the downsteam is geneally teminated by a hydaulic jump. Compaed to M and M pofiles, M 3 cuves ae of elatively shot length.

11 Example 6. : A ectangula channel with a bottom width of 4. m and a bottom slope of.8 has a dischage of.5 m 3 /sec. In a gadually vaied flow in this channel, the depth at a cetain location is found to be.3 m. assuming n.6, detemine the type of GVF pofile. Solution: a) To find the nomal depth y, A By 4y P B + y R A 4y P 4 + y 4 + y b) Citical depth y c, c) Type of pofile, A Q VA A n P.5 q S 5 3 ( 4y ) ( 4 + y ) 3 y 5 3 ( 4 + y ) 3.8 By tial and eo, y.43 m. Q B.5.375m sec q.375 yc. 4m g 9.8 y.43 m > y c.4 m (Mild slope channel, M pofile) Wate suface pofile is of the M type. y.3 m y > y > y c (Region ).5

12 6.4.. S Cuves Figue 6.8 Geneal shapes of S cuves ae given in Fig. (6.8). Asymptotic behavios of each cuve will be examined mathematically. a) S Cuve Wate suface will be in Region fo a steep slope channel and the flow is obviously supecitical. y > y c > y y > y V < V S e < S (S S e ) > y > y c F < ( F ) > S F e (Wate depth will incease in the flow diection) Asymptotic behavio of the wate suface is; > y > y c y, V, F, (- F ) y, V, S e lim y S F e S S

13 The wate suface meets a vey lage depth as a hoizontal asymptote. y y c, F, ( F ) S e S lim y y c F e The wate suface meets the citical depth line vetically. Figue 6.9. S Pofile The S pofile is poduced when the flow fom a steep channel is teminated by a deep pool ceated by an obstuction, such as a wei o dam (Fig. 6.9). At the beginning of the cuve, the flow changes fom the nomal depth (supecitical flow) to subcitical flow though a hydaulic jump. The pofiles extend downsteam with a positive wate slope to each a hoizontal asymptote at the pool elevation. b) S Cuve Wate suface will be in Region fo a steep slope channel and the flow is supecitical. y < y < y c y > y S e < S (S S e ) > y < y c Supecitical flow F > ( F ) < S F e + (Wate depth will decease in the flow diection) 3

14 Asymptotic behavio of the wate suface is; y < y < y c y y c, F, ( F ) S e S lim y y c F e The wate suface meets the citical depth line vetically. y y, V V, S e S, (S S e ) lim y y F The wate suface appoaches the nomal depth asymptotically. Figue 6.. S Pofile Pofiles of the S type occu at the entance egion of a steep channel leading fom a esevoi and at a bake of gade fom mild slopes to steep slope (Fig. 6.). Geneally S pofiles ae shot of length. c) S 3 Cuve Wate suface will be in Region 3 fo a steep slope channel and the flow is supecitical. < y < y y < y S e > S (S S e ) < y < y c Supecitical flow F > ( F ) < S e + F (Wate depth will incease in the flow diection) 4

15 Asymptotic behavio of the wate suface is; < y < y y, S e, (S S e ) - y, F V gy lim (Unknown) y y The angle of the wate suface with the channel bed may be taken as S. y c y y, S e S, (S S e ) 3 lim y y F The wate suface appoaches the nomal depth y asymptotically. Figue 6.. S 3 Pofile Figue 6.. S 3 Pofile Fee flow fom a sluice gate with a steep slope on its downsteam is of the S 3 type (Fig. 6.). The S 3 cuve also esults when a flow exists fom a steepe slope to a less steep slope (Fig. 6.). 5

16 C Cuves Figue 6.3 Geneal shapes of C cuves ae given in Fig. (6.3). Asymptotic behavios of each cuve will be examined mathematically. Since the flow is at citical stage, y y c, thee is no Region. a) C Cuve Wate suface will be in Region fo a citical slope channel. y y c < y < y > y c S e < S (S S e ) > y > y c Subcitical flow F < ( F ) > S F e (Wate depth will incease in the flow diection) Asymptotic behavio of the wate suface is; y, S e, ( S S e ) S y, V F ( F ) lim y S S The wate suface meets a vey lage depth as a hoizontal asymptote. y y c, F ( F ) y y c y, S e S S c (S S e ) lim y y y c (Unknown) C and C 3 pofiles ae vey ae and highly unstable 6

17 H Cuves Figue 6.4 Geneal shapes of H cuves ae given in Fig. (6.4). Fo hoizontal slope channels, unifom flow depth y does not exist. Citical wate depth can be computed fo a given dischage Q and theefoe citical wate depth line can be dawn. Since thee is no unifom wate depth y, Region does not exist. a) H Cuve Wate suface will be in Region fo a hoizontal slope channel. > y > y c y > y c S e < S (S S e ) < y > y c subcitical flow F < ( F ) > + (Wate depth will decease in the flow diection) Asymptotic behavio of the wate suface is; y, S e, ( S S e ) S y, V F ( F ) lim y S S The wate suface meets a vey lage depth as a hoizontal asymptote. y y c, F ( F ) y y c, S e S c S, (S S e ) -S e S e e lim y y c F The wate suface meets the citical depth line vetically. 7

18 b) H 3 Cuve > y > y c y < y c S e > S (S S e ) -S e y y c, supecitical flow, F > ( F ) < S F e + (Wate depth will incease in the flow diection) Asymptotic behavio of the wate suface is; y, S e, (S S e ) - y, F V gy lim (Unknown) y y y c F ( F ) y y c, S e S c S, (S S e ) -S e S lim y y c F e e The wate suface meets the citical depth line vetically. Figue 6.5 A hoizontal channel can be consideed as the lowe limit eached by a mild slope as its bed slope becomes flatte. The H and H 3 pofiles ae simila to M and M 3 pofiles espectively (Fig. 6.5). Howeve, the H cuve has a hoizontal asymptote. 8

19 A Cuves Figue 6.6 Geneal shapes of A cuves ae given in Fig. (6.6). Fo advese slope channels, unifom flow depth y does not exist. Citical wate depth can be computed fo a given dischage Q and theefoe citical wate depth line can be dawn. Since thee is no unifom wate depth y, Region does not exist as well as in A cuves. A and A 3 cuves ae simila to H and H 3 cuves espectively. Figue 6.7 Advese slopes ae athe ae (Fig. 6.7). These pofiles ae of vey shot length Contol Sections A contol section is defined as a section in which a fixed elationship exists between the dischage and depth of flow. Weis, spillways, sluice gates ae some typical examples of stuctues which give ise to contol sections. The citical depth is also a contol point. Howeve, it is effective in a flow pofile which changes fom subcitical to supecitical flow. In the evese case of tansition fom supecitical flow to subcitical flow, a hydaulic jump is usually fomed bypassing the citical depth as a contol point. Any GVF pofile will have at least one contol section. 9

20 (d) Figue 6.8 In the synthesis of GVF pofiles occuing in a seially connected channel elements, the contol sections povide a key to the identification of pope pofile shapes. A few typical contol sections ae shown in Fig. (6.8, a-e). It may be noted that subcitical flows have contols in the downsteam end, while supecitical flows ae govened by contol sections existing at the upsteam end of the channel section. In Figs. (6.8 a and b) fo the M pofile, the contol section (indicated by a dak dot in the figues) is just at the upsteam of the spillway and sluice gate espectively. In Figs. (6.8 b and d) fo M 3 and S 3 pofiles espectively, the contol point is at the vena contacta of the sluice gate flow. In subcitical flow esevoi offtakes (Fig. 6.8 c), even though the dischage is govened

21 by the esevoi elevation, the channel enty section is not stictly a contol section. The wate suface elevation in the channel, will be lowe than the esevoi elevation by a headloss amount equivalent to ( + ζ )V /g whee ζ is the entance loss coefficient. The tue contol section will be at a downsteam location in the channel. Fo the situation shown in Fig. (6.8 c) the citical depth at the fee oveflow at the channel end acts as the downsteam contol. Fo a sudden dop (fee oveflow) due to cuvatue of the steamlines the citical depth usually occus at distance of about 4y c upsteam of the dop. This distance, being small compaed to GVF lengths, is neglected and it is usual to pefom calculations by assuming y c to occu at the dop. (e) Fo a supecitical canal intake (Fig. 6.8 e), the esevoi wate suface falls to the citical depth at the head of the canal and then onwads the wate suface follows the S cuve. The citical depth occuing at the upsteam end of the channel is the contol fo this flow. Figue 6.9

22 6.5.. Feeding a Pool fo Subcitical Flow A mild slope channel dischaging into a pool of vaiable suface elevation is indicated in Fig. (6.9). Fou cases ae shown. ) The pool elevation is highe than the elevation of the nomal (unifom) wate depth line at B. This gives ise to a downing of the channel end. A pofile of the M type is poduced with the pool level at B as contol. ) The pool elevation is lowe than the elevation of the nomal depth line but highe than the citical depth line at B. The pool elevation acts as a contol fo the M cuve. 3) The pool elevation has dopped down to that of the citical depth line at B and the contol is still at the pool elevation. 4) The pool elevation has dopped lowe than the elevation of the citical depth line at B. The wate suface cannot pass though a citical depth at any location othe than B and hence a sudden dop in the wate suface at B is obseved. The citical depth at B is the contol of this flow Feeding a Pool fo Supecitical Flow When the slope of the channel is steep, that is geate than the citical slope, the flow in the channel becomes supecitical. In pactical applications, steep channels ae usually shot, such as aft and log chutes that ae used as spillways. Figue 6. As the contol section in a channel of supecitical flow is at the upsteam end, the delivey of the channel is fully govened by the citical dischage at section.

23 ) When the tailwate level B is less than the outlet depth (nomal depth) at section, the flow in the canal is unaffected by the tailwate. The flow pofile passes though the citical wate depth nea C and appoaches the nomal depth by means of a smooth dawdown cuve of the S type. ) When the tailwate level B is geate than the outlet depth, the tailwate will aise the wate level in the upsteam potion of the canal to fom an S pofile between j and b`, poducing a hydaulic jump at the end j of the pofile. Howeve, the flow upsteam fom the jump will not be affected by the tailwate. Such a hydaulic jump is objectionable and dangeous, paticulaly when the canal is aft chute o some othe stuctue intended to tanspot a floating aft fom the upsteam esevoi to a downsteam pool. A neutalizing each may be suggested fo the solution of this poblem. (Chow, 959). (Fig. 6.) Figue 6.. Elimination of hydaulic jump by neutalizing each. In this each the bottom slope of the channel is made equal to the citical slope. Accoding to a coesponding case of C pofile in Fig. (6.), the tailwate levels will be appoximately hoizontal lines which intesect the suface of flow in the canal without causing any distubance. At the point of intesection, theoetically, thee is a jump of zeo height. 3) As the tailwate ises futhe, the jump will move upsteam, maintaining its height and fom in the unifom flow zone nb, until it eaches point n. The height of the jump becomes zeo when it eaches the citical depth at c. In the meantime, the flow pofile eaches its theoetical limit cb`` of the S pofile. Beyond this limit the incoming flow will be diectly affected by the tailwate. In pactical applications, the hoizontal line cb``` may be taken as the pactical limit of the tailwate stage. 3

24 6.6. Analysis of Flow Pofile A channel caying a gadually vaied flow can in geneal contain diffeent pismoidal channel coss-sections of vaying hydaulic popeties. Thee can be a numbe of contol sections at vaious locations. To detemine the esulting wate suface pofile in a given case, one should be in a position to analyze the effects of vaious channel sections and contols connected in seies. Simple cases ae illustated to povide infomation and expeience to handle moe complex cases Beak in Gades Simple situations of a seies combination of two channel sections with diffeing bed slopes ae consideed. In Fig. (6..a), a beak in gade fom a mild channel to a milde channel is shown. It is necessay to fist daw the citical-depth line (CDL) and the nomal-depth line (NDL) fo both slopes. Since y c does not depend upon the slope fo a taken Q dischage, the CDL is at a constant height above the channel bed in both slopes. The nomal depth y fo the mild slope is lowe than that of the of the milde slope (y ). In this case, y acts as a contol, simila to the wei o spillway case and an M backwate cuve is poduced in the mild slope channel. Vaious combinations of slopes and the esulting GVF pofiles ae pesented in Fig. ( 6., a-h). It may be noted that in some situations thee can be moe than one possible pofiles. Fo example, in Fig. (6. e), a jump and S pofile o an M 3 pofile and a jump possible. The paticula cuve in this case depends on the channel and its flow popeties. In the examples indicated in Fig. (6.), the section whee the gade changes acts a contol section and this can be classified as a natual contol. It should be noted that even though the bed slope is consideed as the only vaiable in the above examples, the same type of analysis would hold good fo channel sections in which thee is a maked change in the oughness chaacteistics with o without change in the bed slope. A long each of unlined canal followed by a line each seves as a typical example fo the same. A change in the channel geomety (the bed width o side slope) beyond a section while etaining the pismoidal natue in each each also leads to a natual contol section. 4

25 5

26 6

27 Figue Seial Combination of Channel Sections To analyze a geneal poblem of many channel sections and contols, the following steps ae to be applied.. Daw the longitudinal section of the system.. Calculate the citical depth and nomal depths of vaious eaches and daw the CDL and NDL in all eaches. 3. Mak all the contols, both the imposed as well as natual contols. 4. Identify the possible pofiles. Example 6. : Identify and sketch the GVF pofiles in thee mild slopes which could be descibed as mild, steepe mild and milde. The thee slopes ae in seies. The last slope has a sluice gate in the middle of the each and the downsteam end of the channel has a fee ovefall. Solution: The longitudinal pofile of the channel, citical depth line and nomal depth lines fo the vaious eaches ae shown in the Fig. (6.3). The fee ovefall at E is obviously a contol. The vena contacta downsteam of the sluice gate at D is anothe contol. Since fo subcitical flow the contol is at the downsteam end of the channel, the highe of the two nomal depths at C acts as a contol fo the each CB, giving ise to an M pofile ove CB. At B, the nomal depth of the channel CB acts as a contol giving ise to an M pofile ove AB. The contols ae maked distinctly in Fig. (6.3). With these contols the possible flow pofiles ae: an M pofile on channel AB, M pofile on channel BC, M 3 pofile and M pofile though a jump on the stetch DE. 7

28 Figue 6.3 Example 6.3: A tapezoidal channel has thee eaches A, B, and C connected in seies with the following physical chaacteistics. Reach 3 Bed width (B) 4. m 4. m 4. m Side slope (m)... Bed slope S n.5..5 Fo a dischage Q.5 m 3 /sec though this channel, sketch the esulting wate suface pofiles. The length of the eaches can be assumed to be sufficiently long fo the GVF pofiles to develop fully. Solution: The nomal depths and citical wate depths in the vaious eaches ae calculated as: T y m B4m A ( B + my) P B + y T B + my y + m 8

29 A (4 + y) y P 4 + y T 4 + y y Q AV.5 A A n P [( 4 + y ) y ] n 3 S S.5 ( y ) 3 Unifom flow depths fo the given data fo evey each ae calculated by tial and eo method; Reach A: Reach B: Reach C: S A.4, n A.5 y A.6 m S B.9, n B. y B.8 m S C.4, n C.5 y C.7 m Since the channel is pismoidal (the geomety does not change in the eaches), thee will be only one citical wate depth. Q T ga c 3 c ( 4 + yc ) [( 4 + y ) y ] c c 3 y c.3 m Reach A is a mild slope channel as y A.6 m > y c.3 m and the flow is subcitical. Reach B and C ae steep slope channels as y B, y C < y c and the flow is supecitical on both eaches. Reach B is steepe than each C. The vaious eaches ae schematically shown in Fig. (6.4). The CDL is dawn at a height of.3 m above the bed level and NDLs ae dawn at the calculated y values. The contols ae maked in the figue. Reach A will have an M dawdown cuve, each B an S dawdown cuve and each C a ising cuve as shown in the figue. It may be noted that the esulting pofile as above is a seial combination of Fig. (6. d and f). 9

30 Figue 6.3 3

### VISCOSITY OF BIO-DIESEL FUELS

VISCOSITY OF BIO-DIESEL FUELS One of the key assumptions fo ideal gases is that the motion of a given paticle is independent of any othe paticles in the system. With this assumption in place, one can use

### Problem Set # 9 Solutions

Poblem Set # 9 Solutions Chapte 12 #2 a. The invention of the new high-speed chip inceases investment demand, which shifts the cuve out. That is, at evey inteest ate, fims want to invest moe. The incease

### An Introduction to Omega

An Intoduction to Omega Con Keating and William F. Shadwick These distibutions have the same mean and vaiance. Ae you indiffeent to thei isk-ewad chaacteistics? The Finance Development Cente 2002 1 Fom

### Chapter 3 Savings, Present Value and Ricardian Equivalence

Chapte 3 Savings, Pesent Value and Ricadian Equivalence Chapte Oveview In the pevious chapte we studied the decision of households to supply hous to the labo maket. This decision was a static decision,

### Lab #7: Energy Conservation

Lab #7: Enegy Consevation Photo by Kallin http://www.bungeezone.com/pics/kallin.shtml Reading Assignment: Chapte 7 Sections 1,, 3, 5, 6 Chapte 8 Sections 1-4 Intoduction: Pehaps one of the most unusual

### FXA 2008. Candidates should be able to : Describe how a mass creates a gravitational field in the space around it.

Candidates should be able to : Descibe how a mass ceates a gavitational field in the space aound it. Define gavitational field stength as foce pe unit mass. Define and use the peiod of an object descibing

### Experiment 6: Centripetal Force

Name Section Date Intoduction Expeiment 6: Centipetal oce This expeiment is concened with the foce necessay to keep an object moving in a constant cicula path. Accoding to Newton s fist law of motion thee

### Spirotechnics! September 7, 2011. Amanda Zeringue, Michael Spannuth and Amanda Zeringue Dierential Geometry Project

Spiotechnics! Septembe 7, 2011 Amanda Zeingue, Michael Spannuth and Amanda Zeingue Dieential Geomety Poject 1 The Beginning The geneal consensus of ou goup began with one thought: Spiogaphs ae awesome.

### Revision Guide for Chapter 11

Revision Guide fo Chapte 11 Contents Student s Checklist Revision Notes Momentum... 4 Newton's laws of motion... 4 Gavitational field... 5 Gavitational potential... 6 Motion in a cicle... 7 Summay Diagams

### 1.4 Phase Line and Bifurcation Diag

Dynamical Systems: Pat 2 2 Bifucation Theoy In pactical applications that involve diffeential equations it vey often happens that the diffeential equation contains paametes and the value of these paametes

### 12. Rolling, Torque, and Angular Momentum

12. olling, Toque, and Angula Momentum 1 olling Motion: A motion that is a combination of otational and tanslational motion, e.g. a wheel olling down the oad. Will only conside olling with out slipping.

### Converting knowledge Into Practice

Conveting knowledge Into Pactice Boke Nightmae srs Tend Ride By Vladimi Ribakov Ceato of Pips Caie 20 of June 2010 2 0 1 0 C o p y i g h t s V l a d i m i R i b a k o v 1 Disclaime and Risk Wanings Tading

### Carter-Penrose diagrams and black holes

Cate-Penose diagams and black holes Ewa Felinska The basic intoduction to the method of building Penose diagams has been pesented, stating with obtaining a Penose diagam fom Minkowski space. An example

### Deflection of Electrons by Electric and Magnetic Fields

Physics 233 Expeiment 42 Deflection of Electons by Electic and Magnetic Fields Refeences Loain, P. and D.R. Coson, Electomagnetism, Pinciples and Applications, 2nd ed., W.H. Feeman, 199. Intoduction An

### NUCLEAR MAGNETIC RESONANCE

19 Jul 04 NMR.1 NUCLEAR MAGNETIC RESONANCE In this expeiment the phenomenon of nuclea magnetic esonance will be used as the basis fo a method to accuately measue magnetic field stength, and to study magnetic

### Introduction to Electric Potential

Univesiti Teknologi MARA Fakulti Sains Gunaan Intoduction to Electic Potential : A Physical Science Activity Name: HP: Lab # 3: The goal of today s activity is fo you to exploe and descibe the electic

### 8-1 Newton s Law of Universal Gravitation

8-1 Newton s Law of Univesal Gavitation One of the most famous stoies of all time is the stoy of Isaac Newton sitting unde an apple tee and being hit on the head by a falling apple. It was this event,

### Gauss Law. Physics 231 Lecture 2-1

Gauss Law Physics 31 Lectue -1 lectic Field Lines The numbe of field lines, also known as lines of foce, ae elated to stength of the electic field Moe appopiately it is the numbe of field lines cossing

### The Binomial Distribution

The Binomial Distibution A. It would be vey tedious if, evey time we had a slightly diffeent poblem, we had to detemine the pobability distibutions fom scatch. Luckily, thee ae enough similaities between

### Physics 235 Chapter 5. Chapter 5 Gravitation

Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

### 2 - ELECTROSTATIC POTENTIAL AND CAPACITANCE Page 1

- ELECTROSTATIC POTENTIAL AND CAPACITANCE Page. Line Integal of Electic Field If a unit positive chage is displaced by `given by dw E. dl dl in an electic field of intensity E, wok done is Line integation

### INVESTIGATION OF FLOW INSIDE AN AXIAL-FLOW PUMP OF GV IMP TYPE

1 INVESTIGATION OF FLOW INSIDE AN AXIAL-FLOW PUMP OF GV IMP TYPE ANATOLIY A. YEVTUSHENKO 1, ALEXEY N. KOCHEVSKY 1, NATALYA A. FEDOTOVA 1, ALEXANDER Y. SCHELYAEV 2, VLADIMIR N. KONSHIN 2 1 Depatment of

### Figure 2. So it is very likely that the Babylonians attributed 60 units to each side of the hexagon. Its resulting perimeter would then be 360!

1. What ae angles? Last time, we looked at how the Geeks intepeted measument of lengths. Howeve, as fascinated as they wee with geomety, thee was a shape that was much moe enticing than any othe : the

### Coordinate Systems L. M. Kalnins, March 2009

Coodinate Sstems L. M. Kalnins, Mach 2009 Pupose of a Coodinate Sstem The pupose of a coodinate sstem is to uniquel detemine the position of an object o data point in space. B space we ma liteall mean

### Introduction to Fluid Mechanics

Chapte 1 1 1.6. Solved Examples Example 1.1 Dimensions and Units A body weighs 1 Ibf when exposed to a standad eath gavity g = 3.174 ft/s. (a) What is its mass in kg? (b) What will the weight of this body

### LINES AND TANGENTS IN POLAR COORDINATES

LINES AND TANGENTS IN POLAR COORDINATES ROGER ALEXANDER DEPARTMENT OF MATHEMATICS 1. Pola-coodinate equations fo lines A pola coodinate system in the plane is detemined by a point P, called the pole, and

### Questions & Answers Chapter 10 Software Reliability Prediction, Allocation and Demonstration Testing

M13914 Questions & Answes Chapte 10 Softwae Reliability Pediction, Allocation and Demonstation Testing 1. Homewok: How to deive the fomula of failue ate estimate. λ = χ α,+ t When the failue times follow

### Chapter 13 Gravitation. Problems: 1, 4, 5, 7, 18, 19, 25, 29, 31, 33, 43

Chapte 13 Gavitation Poblems: 1, 4, 5, 7, 18, 19, 5, 9, 31, 33, 43 Evey object in the univese attacts evey othe object. This is called gavitation. We e use to dealing with falling bodies nea the Eath.

### (a) The centripetal acceleration of a point on the equator of the Earth is given by v2. The velocity of the earth can be found by taking the ratio of

Homewok VI Ch. 7 - Poblems 15, 19, 22, 25, 35, 43, 51. Poblem 15 (a) The centipetal acceleation of a point on the equato of the Eath is given by v2. The velocity of the eath can be found by taking the

### Algebra and Trig. I. A point is a location or position that has no size or dimension.

Algeba and Tig. I 4.1 Angles and Radian Measues A Point A A B Line AB AB A point is a location o position that has no size o dimension. A line extends indefinitely in both diections and contains an infinite

### Problems of the 2 nd International Physics Olympiads (Budapest, Hungary, 1968)

Poblems of the nd ntenational Physics Olympiads (Budapest Hungay 968) Péte Vankó nstitute of Physics Budapest Univesity of Technical Engineeing Budapest Hungay Abstact Afte a shot intoduction the poblems

### 2 r2 θ = r2 t. (3.59) The equal area law is the statement that the term in parentheses,

3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects

### The Role of Gravity in Orbital Motion

! The Role of Gavity in Obital Motion Pat of: Inquiy Science with Datmouth Developed by: Chistophe Caoll, Depatment of Physics & Astonomy, Datmouth College Adapted fom: How Gavity Affects Obits (Ohio State

### CHAPTER 10 Aggregate Demand I

CHAPTR 10 Aggegate Demand I Questions fo Review 1. The Keynesian coss tells us that fiscal policy has a multiplied effect on income. The eason is that accoding to the consumption function, highe income

DYNAMIS AND STRUTURAL LOADING IN WIND TURBINES M. Ragheb 12/30/2008 INTRODUTION The loading egimes to which wind tubines ae subject to ae extemely complex equiing special attention in thei design, opeation

### Fluids Lecture 15 Notes

Fluids Lectue 15 Notes 1. Unifom flow, Souces, Sinks, Doublets Reading: Andeson 3.9 3.12 Unifom Flow Definition A unifom flow consists of a velocit field whee V = uî + vĵ is a constant. In 2-D, this velocit

### Instituto Superior Técnico Av. Rovisco Pais, 1 1049-001 Lisboa E-mail: virginia.infante@ist.utl.pt

FATIGUE LIFE TIME PREDICTIO OF POAF EPSILO TB-30 AIRCRAFT - PART I: IMPLEMETATIO OF DIFERET CYCLE COUTIG METHODS TO PREDICT THE ACCUMULATED DAMAGE B. A. S. Seano 1, V. I. M.. Infante 2, B. S. D. Maado

### ON THE (Q, R) POLICY IN PRODUCTION-INVENTORY SYSTEMS

ON THE R POLICY IN PRODUCTION-INVENTORY SYSTEMS Saifallah Benjaafa and Joon-Seok Kim Depatment of Mechanical Engineeing Univesity of Minnesota Minneapolis MN 55455 Abstact We conside a poduction-inventoy

### UNIT CIRCLE TRIGONOMETRY

UNIT CIRCLE TRIGONOMETRY The Unit Cicle is the cicle centeed at the oigin with adius unit (hence, the unit cicle. The equation of this cicle is + =. A diagam of the unit cicle is shown below: + = - - -

### Episode 401: Newton s law of universal gravitation

Episode 401: Newton s law of univesal gavitation This episode intoduces Newton s law of univesal gavitation fo point masses, and fo spheical masses, and gets students pactising calculations of the foce

### Comparing Availability of Various Rack Power Redundancy Configurations

Compaing Availability of Vaious Rack Powe Redundancy Configuations By Victo Avela White Pape #48 Executive Summay Tansfe switches and dual-path powe distibution to IT equipment ae used to enhance the availability

### STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION

Page 1 STUDENT RESPONSE TO ANNUITY FORMULA DERIVATION C. Alan Blaylock, Hendeson State Univesity ABSTRACT This pape pesents an intuitive appoach to deiving annuity fomulas fo classoom use and attempts

### AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM

AN IMPLEMENTATION OF BINARY AND FLOATING POINT CHROMOSOME REPRESENTATION IN GENETIC ALGORITHM Main Golub Faculty of Electical Engineeing and Computing, Univesity of Zageb Depatment of Electonics, Micoelectonics,

### Experiment MF Magnetic Force

Expeiment MF Magnetic Foce Intoduction The magnetic foce on a cuent-caying conducto is basic to evey electic moto -- tuning the hands of electic watches and clocks, tanspoting tape in Walkmans, stating

### Lecture 16: Color and Intensity. and he made him a coat of many colours. Genesis 37:3

Lectue 16: Colo and Intensity and he made him a coat of many colous. Genesis 37:3 1. Intoduction To display a pictue using Compute Gaphics, we need to compute the colo and intensity of the light at each

### 1240 ev nm 2.5 ev. (4) r 2 or mv 2 = ke2

Chapte 5 Example The helium atom has 2 electonic enegy levels: E 3p = 23.1 ev and E 2s = 20.6 ev whee the gound state is E = 0. If an electon makes a tansition fom 3p to 2s, what is the wavelength of the

### Supplementary Material for EpiDiff

Supplementay Mateial fo EpiDiff Supplementay Text S1. Pocessing of aw chomatin modification data In ode to obtain the chomatin modification levels in each of the egions submitted by the use QDCMR module

### dz + η 1 r r 2 + c 1 ln r + c 2 subject to the boundary conditions of no-slip side walls and finite force over the fluid length u z at r = 0

Poiseuille Flow Jean Louis Maie Poiseuille, a Fench physicist and physiologist, was inteested in human blood flow and aound 1840 he expeimentally deived a law fo flow though cylindical pipes. It s extemely

### Problem Set 6: Solutions

UNIVESITY OF ALABAMA Depatment of Physics and Astonomy PH 16-4 / LeClai Fall 28 Poblem Set 6: Solutions 1. Seway 29.55 Potons having a kinetic enegy of 5. MeV ae moving in the positive x diection and ente

### Comparing Availability of Various Rack Power Redundancy Configurations

Compaing Availability of Vaious Rack Powe Redundancy Configuations White Pape 48 Revision by Victo Avela > Executive summay Tansfe switches and dual-path powe distibution to IT equipment ae used to enhance

### Semipartial (Part) and Partial Correlation

Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated

### Software Engineering and Development

I T H E A 67 Softwae Engineeing and Development SOFTWARE DEVELOPMENT PROCESS DYNAMICS MODELING AS STATE MACHINE Leonid Lyubchyk, Vasyl Soloshchuk Abstact: Softwae development pocess modeling is gaining

### Vector Calculus: Are you ready? Vectors in 2D and 3D Space: Review

Vecto Calculus: Ae you eady? Vectos in D and 3D Space: Review Pupose: Make cetain that you can define, and use in context, vecto tems, concepts and fomulas listed below: Section 7.-7. find the vecto defined

### Voltage ( = Electric Potential )

V-1 Voltage ( = Electic Potential ) An electic chage altes the space aound it. Thoughout the space aound evey chage is a vecto thing called the electic field. Also filling the space aound evey chage is

### YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH

nd INTERNATIONAL TEXTILE, CLOTHING & ESIGN CONFERENCE Magic Wold of Textiles Octobe 03 d to 06 th 004, UBROVNIK, CROATIA YARN PROPERTIES MEASUREMENT: AN OPTICAL APPROACH Jana VOBOROVA; Ashish GARG; Bohuslav

### PHYSICS 111 HOMEWORK SOLUTION #5. March 3, 2013

PHYSICS 111 HOMEWORK SOLUTION #5 Mach 3, 2013 0.1 You 3.80-kg physics book is placed next to you on the hoizontal seat of you ca. The coefficient of static fiction between the book and the seat is 0.650,

### Gravitation. AP Physics C

Gavitation AP Physics C Newton s Law of Gavitation What causes YOU to be pulled down? THE EARTH.o moe specifically the EARTH S MASS. Anything that has MASS has a gavitational pull towads it. F α Mm g What

### Performance Analysis of an Inverse Notch Filter and Its Application to F 0 Estimation

Cicuits and Systems, 013, 4, 117-1 http://dx.doi.og/10.436/cs.013.41017 Published Online Januay 013 (http://www.scip.og/jounal/cs) Pefomance Analysis of an Invese Notch Filte and Its Application to F 0

### Ilona V. Tregub, ScD., Professor

Investment Potfolio Fomation fo the Pension Fund of Russia Ilona V. egub, ScD., Pofesso Mathematical Modeling of Economic Pocesses Depatment he Financial Univesity unde the Govenment of the Russian Fedeation

### Chapter 4: Fluid Kinematics

Oveview Fluid kinematics deals with the motion of fluids without consideing the foces and moments which ceate the motion. Items discussed in this Chapte. Mateial deivative and its elationship to Lagangian

### 9.5 Volume of Pyramids

Page of 7 9.5 Volume of Pyamids and Cones Goal Find the volumes of pyamids and cones. Key Wods pyamid p. 49 cone p. 49 volume p. 500 In the puzzle below, you can see that the squae pism can be made using

### est using the formula I = Prt, where I is the interest earned, P is the principal, r is the interest rate, and t is the time in years.

9.2 Inteest Objectives 1. Undestand the simple inteest fomula. 2. Use the compound inteest fomula to find futue value. 3. Solve the compound inteest fomula fo diffeent unknowns, such as the pesent value,

### The Electric Potential, Electric Potential Energy and Energy Conservation. V = U/q 0. V = U/q 0 = -W/q 0 1V [Volt] =1 Nm/C

Geneal Physics - PH Winte 6 Bjoen Seipel The Electic Potential, Electic Potential Enegy and Enegy Consevation Electic Potential Enegy U is the enegy of a chaged object in an extenal electic field (Unit

### A Review of Concentric Annular Heat Pipes

Heat Tansfe Engineeing, 266):45 58, 2005 Copyight C Taylo and Fancis Inc. ISSN: 0145-7632 pint / 1521-0537 online DOI: 10.1080/01457630590950934 A Review of Concentic Annula Heat Pipes A. NOURI-BORUJERDI

### Physics HSC Course Stage 6. Space. Part 1: Earth s gravitational field

Physics HSC Couse Stage 6 Space Pat 1: Eath s gavitational field Contents Intoduction... Weight... 4 The value of g... 7 Measuing g...8 Vaiations in g...11 Calculating g and W...13 You weight on othe

### In the lecture on double integrals over non-rectangular domains we used to demonstrate the basic idea

Double Integals in Pola Coodinates In the lectue on double integals ove non-ectangula domains we used to demonstate the basic idea with gaphics and animations the following: Howeve this paticula example

### Open Economies. Chapter 32. A Macroeconomic Theory of the Open Economy. Basic Assumptions of a Macroeconomic Model of an Open Economy

Chapte 32. A Macoeconomic Theoy of the Open Economy Open Economies An open economy is one that inteacts feely with othe economies aound the wold. slide 0 slide 1 Key Macoeconomic Vaiables in an Open Economy

### Chapter 30: Magnetic Fields Due to Currents

d Chapte 3: Magnetic Field Due to Cuent A moving electic chage ceate a magnetic field. One of the moe pactical way of geneating a lage magnetic field (.1-1 T) i to ue a lage cuent flowing though a wie.

### Forces & Magnetic Dipoles. r r τ = μ B r

Foces & Magnetic Dipoles x θ F θ F. = AI τ = U = Fist electic moto invented by Faaday, 1821 Wie with cuent flow (in cup of Hg) otates aound a a magnet Faaday s moto Wie with cuent otates aound a Pemanent

### Proc. Int. Joint Conf. On Neural Networks, (1):587{592, San Diego, June Non-linear Prediction with Self-organizing Maps

Poc. Int. Joint Conf. On Neual Netwoks, (1):587{592, San Diego, June 1990 Non-linea Pediction with Self-oganizing Maps Jog Walte, Helge Ritte and Klaus Schulten Beckman-Institute and Depatment of Physics

### Database Management Systems

Contents Database Management Systems (COP 5725) D. Makus Schneide Depatment of Compute & Infomation Science & Engineeing (CISE) Database Systems Reseach & Development Cente Couse Syllabus 1 Sping 2012

### Magnetic Bearing with Radial Magnetized Permanent Magnets

Wold Applied Sciences Jounal 23 (4): 495-499, 2013 ISSN 1818-4952 IDOSI Publications, 2013 DOI: 10.5829/idosi.wasj.2013.23.04.23080 Magnetic eaing with Radial Magnetized Pemanent Magnets Vyacheslav Evgenevich

### Problems of the 2 nd and 9 th International Physics Olympiads (Budapest, Hungary, 1968 and 1976)

Poblems of the nd and 9 th Intenational Physics Olympiads (Budapest Hungay 968 and 976) Péte Vankó Institute of Physics Budapest Univesity of Technology and Economics Budapest Hungay Abstact Afte a shot

### EXPERIENCE OF USING A CFD CODE FOR ESTIMATING THE NOISE GENERATED BY GUSTS ALONG THE SUN- ROOF OF A CAR

EXPERIENCE OF USING A CFD CODE FOR ESTIMATING THE NOISE GENERATED BY GUSTS ALONG THE SUN- ROOF OF A CAR Liang S. Lai* 1, Geogi S. Djambazov 1, Choi -H. Lai 1, Koulis A. Peicleous 1, and Fédéic Magoulès

### TECHNICAL DATA. JIS (Japanese Industrial Standard) Screw Thread. Specifications

JIS (Japanese Industial Standad) Scew Thead Specifications TECNICAL DATA Note: Although these specifications ae based on JIS they also apply to and DIN s. Some comments added by Mayland Metics Coutesy

### Structure and evolution of circumstellar disks during the early phase of accretion from a parent cloud

Cente fo Tubulence Reseach Annual Reseach Biefs 2001 209 Stuctue and evolution of cicumstella disks duing the ealy phase of accetion fom a paent cloud By Olusola C. Idowu 1. Motivation and Backgound The

### Efficient Redundancy Techniques for Latency Reduction in Cloud Systems

Efficient Redundancy Techniques fo Latency Reduction in Cloud Systems 1 Gaui Joshi, Emina Soljanin, and Gegoy Wonell Abstact In cloud computing systems, assigning a task to multiple seves and waiting fo

### An Epidemic Model of Mobile Phone Virus

An Epidemic Model of Mobile Phone Vius Hui Zheng, Dong Li, Zhuo Gao 3 Netwok Reseach Cente, Tsinghua Univesity, P. R. China zh@tsinghua.edu.cn School of Compute Science and Technology, Huazhong Univesity

### Financing Terms in the EOQ Model

Financing Tems in the EOQ Model Habone W. Stuat, J. Columbia Business School New Yok, NY 1007 hws7@columbia.edu August 6, 004 1 Intoduction This note discusses two tems that ae often omitted fom the standad

### Solution Derivations for Capa #8

Solution Deivations fo Capa #8 1) A ass spectoete applies a voltage of 2.00 kv to acceleate a singly chaged ion (+e). A 0.400 T field then bends the ion into a cicula path of adius 0.305. What is the ass

### The transport performance evaluation system building of logistics enterprises

Jounal of Industial Engineeing and Management JIEM, 213 6(4): 194-114 Online ISSN: 213-953 Pint ISSN: 213-8423 http://dx.doi.og/1.3926/jiem.784 The tanspot pefomance evaluation system building of logistics

### Hour Exam No.1. p 1 v. p = e 0 + v^b. Note that the probe is moving in the direction of the unit vector ^b so the velocity vector is just ~v = v^b and

Hou Exam No. Please attempt all of the following poblems befoe the due date. All poblems count the same even though some ae moe complex than othes. Assume that c units ae used thoughout. Poblem A photon

### Japan s trading losses reach JPY20 trillion

IEEJ: Mach 2014. All Rights Reseved. Japan s tading losses each JPY20 tillion Enegy accounts fo moe than half of the tading losses YANAGISAWA Akia Senio Economist Enegy Demand, Supply and Foecast Goup

### Magnetic Field and Magnetic Forces. Young and Freedman Chapter 27

Magnetic Field and Magnetic Foces Young and Feedman Chapte 27 Intoduction Reiew - electic fields 1) A chage (o collection of chages) poduces an electic field in the space aound it. 2) The electic field

### Chapter 22. Outside a uniformly charged sphere, the field looks like that of a point charge at the center of the sphere.

Chapte.3 What is the magnitude of a point chage whose electic field 5 cm away has the magnitude of.n/c. E E 5.56 1 11 C.5 An atom of plutonium-39 has a nuclea adius of 6.64 fm and atomic numbe Z94. Assuming

### ESCAPE VELOCITY EXAMPLES

ESCAPE VELOCITY EXAMPLES 1. Escape velocity is the speed that an object needs to be taveling to beak fee of planet o moon's gavity and ente obit. Fo example, a spacecaft leaving the suface of Eath needs

### Avoided emissions kgco 2eq /m 2 Reflecting surfaces 130

ANALYSIS OF GLOBAL WARMING MITIGATION BY WHITE REFLECTING SURFACES Fedeico Rossi, Andea Nicolini Univesity of Peugia, CIRIAF Via G.Duanti 67 0615 Peugia, Italy T: +9-075-585846; F: +9-075-5848470; E: fossi@unipg.it

### SELF-INDUCTANCE AND INDUCTORS

MISN-0-144 SELF-INDUCTANCE AND INDUCTORS SELF-INDUCTANCE AND INDUCTORS by Pete Signell Michigan State Univesity 1. Intoduction.............................................. 1 A 2. Self-Inductance L.........................................

### Determining solar characteristics using planetary data

Detemining sola chaacteistics using planetay data Intoduction The Sun is a G type main sequence sta at the cente of the Sola System aound which the planets, including ou Eath, obit. In this inestigation

### Trading Volume and Serial Correlation in Stock Returns in Pakistan. Abstract

Tading Volume and Seial Coelation in Stock Retuns in Pakistan Khalid Mustafa Assistant Pofesso Depatment of Economics, Univesity of Kaachi e-mail: khalidku@yahoo.com and Mohammed Nishat Pofesso and Chaiman,

### INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS

INITIAL MARGIN CALCULATION ON DERIVATIVE MARKETS OPTION VALUATION FORMULAS Vesion:.0 Date: June 0 Disclaime This document is solely intended as infomation fo cleaing membes and othes who ae inteested in

### Thank you for participating in Teach It First!

Thank you fo paticipating in Teach It Fist! This Teach It Fist Kit contains a Common Coe Suppot Coach, Foundational Mathematics teache lesson followed by the coesponding student lesson. We ae confident

### MATHEMATICAL SIMULATION OF MASS SPECTRUM

MATHEMATICA SIMUATION OF MASS SPECTUM.Beánek, J.Knížek, Z. Pulpán 3, M. Hubálek 4, V. Novák Univesity of South Bohemia, Ceske Budejovice, Chales Univesity, Hadec Kalove, 3 Univesity of Hadec Kalove, Hadec

### The LCOE is defined as the energy price (\$ per unit of energy output) for which the Net Present Value of the investment is zero.

Poject Decision Metics: Levelized Cost of Enegy (LCOE) Let s etun to ou wind powe and natual gas powe plant example fom ealie in this lesson. Suppose that both powe plants wee selling electicity into the

### Problems on Force Exerted by a Magnetic Fields from Ch 26 T&M

Poblems on oce Exeted by a Magnetic ields fom Ch 6 TM Poblem 6.7 A cuent-caying wie is bent into a semicicula loop of adius that lies in the xy plane. Thee is a unifom magnetic field B Bk pependicula to

### PY1052 Problem Set 8 Autumn 2004 Solutions

PY052 Poblem Set 8 Autumn 2004 Solutions H h () A solid ball stats fom est at the uppe end of the tack shown and olls without slipping until it olls off the ight-hand end. If H 6.0 m and h 2.0 m, what

### A Capacitated Commodity Trading Model with Market Power

A Capacitated Commodity Tading Model with Maket Powe Victo Matínez-de-Albéniz Josep Maia Vendell Simón IESE Business School, Univesity of Navaa, Av. Peason 1, 08034 Bacelona, Spain VAlbeniz@iese.edu JMVendell@iese.edu

xam #1 Review Answes 1. Given the following pobability distibution, calculate the expected etun, vaiance and standad deviation fo Secuity J. State Pob (R) 1 0.2 10% 2 0.6 15 3 0.2 20 xpected etun = 0.2*10%