Queens School Physics Department

Transcription

1 Queens School Physics Deprtment S Bridging Workbook Nme:

2 Chpter : Rerrnging equtions The first step in lerning to mnipulte n eqution is your bility to see how it is done once nd then repet the process gin nd gin until it becomes second nture to you. In order to show the process once I will be using letters rther thn physicl concepts. You cn rerrnge n eqution bc with b s the subject b c or c s the subject c b ny of these three symbols b, c, cn be itself summtion, subtrction, multipliction, division, or combintion of ll. So, when you see more complicted eqution, try to identify its three individul prts, b, c before you strt rerrnging it. Worked exmples v v Eqution First Rerrngement Second Rerrngement f v f T T f f v u f u f v f v f T u f THINK! s you cn see from the third worked exmple, not ll rerrngements re useful. In fct, for the lens eqution only the second rerrngement cn be useful in problems. So, in order to improve your criticl thinking nd know which rerrngement is the most useful in every sitution, you must prctise with s mny equtions s you cn.

3 NOW TRY THIS! From now on the multipliction sign will not be shown, so written s bc bc Fill in the blnk spces with your own exmples. will be simply Eqution First Rerrngement Second Rerrngement (Power of lens) P f f v (Mgnifiction of lens) m u v u c (refrctive index) n v c v Q (current) I (electric potentil) (power) t E P t E V Q (power) P VI (conductnce) (resistnce) (resistnce) G V R I R G I V (power) P I R (power) (stress) V P R F F (strin) x l x l (Young s modulus) E (conductnce) G L (resistnce) R L (resistivity) 3

4 (phse ngle) (displcement) ft y sin f t sin (Young s interference) (quntum energy) E hf (electron wvelength) h mv L x d f h 4

5 5 Chpter : Rtios. Mnipulting rtios These re the most useful wys of mnipulting rtios. They will help you when you rerrnge equtions. First: C B D D C B Second: C D B D C B Third: D D C B B D C B Fourth: D D C B B D C B nd finlly, this is how you simplify complex frction: c b d d c b THINK! It is lwys useful to convert complex frction into simple one, using the form bove, before you try nything else. If b or d re missing, substitute them with number. NOW TRY THIS! Here re some exmples to do: [6] 3 4

6 Molecules of DN hve been stretched using opticl tweezers. Using the following dt find the cross sectionl re of DN strnd, if the Young s modulus of DN is 0 8 P nd lod of 4 x 0-0 N results in 0% strin. [] Here is how you strt: mnipulte the eqution until you hve on the left-hnd side nd only t this point substitute ll the symbols for their equivlent numbers. E E F (ns: x0-7 m ) 6

7 . How to convert sentences into rtios Hve look t these problems nd try to solve them. Problem nn hs 800 pples in bskets. Ech bsket holds 6 pples. How mny bskets does she hve? (ns: 50 bskets) Problem The electric current through wire is 3 (3 Coulombs per second). Ech electron hs chrge of.6 x 0-9 C. How mny electrons pss through the wire per second? (ns:.88 x 0 9 electrons) Problem 3 swimming pool is 7 m long. trnsverse wve of 3 m wvelength is creted. How mny complete oscilltions re creted if the wve fills the pool? (ns: 9 wves) THINK! Did you find problem more difficult to do? If yes, think bout this: ll the problems follow the sme logicl sequence. Bskets (in problem ) correspond to electrons (in problem ) nd wvelengths (in problem 3). pples (in problem ) correspond to chrge (in problem ) nd length (in problem 3). Here is nother set of problems: 7

8 Problem 4 John hs 47 pers in bskets. How mny bskets does he need for 4 pers? (ns: bskets) Problem 5 The weight of 50.0 kg person on the moon is 80.0N. How much would 7.0 kg person weigh on the moon? (ns: 5 N) Problem 6 When stereo sound informtion is trnsmitted through cble, 3 bits re sent every.7μs. Clculte how mny bits you cn send during seconds ( s = x 0 6 μs) (ns:.8 x 0 6 bits) THINK! Did you find problem 6 more difficult to do? If yes, think bout this: ll the problems follow the sme logicl sequence. Bskets (in problem 4) correspond to weight (in problem 5) nd bits (in problem 6). Pers (in problem 4) correspond to mss (in problem 5) nd time (in problem 6). FOLLOW THIS! You cn solve ll the bove problems nd mny more using the method below: STEP. Write two sentences, one exctly below the other, tht describe the reltionship between the two vribles. Mke sure you hve the sme vrible on ech side. Use the letter x for the unknown vrible. e.g. 8 bits in byte 300 bits in x byte 8

9 STEP. Convert the two sentences into rtio e.g x STEP 3. Use the properties of rtios to help you rerrnge nd solve the problem e.g. 8x 300 x x 400 NOW TRY THIS! Convert the degrees to rdins nd the rdins to degrees using rtios: e.g. 80 degrees in π rdins x degrees in y rdins Degrees Rdins 4 6 9

10 Chpter 3: How to use nd convert prefixes Mthemticl Prefixes [] Prefix Symbol Nme Multiplier femto f qudrillionth 0-5 pico p trillionth 0 - nno n billionth 0-9 micro µ millionth 0-6 milli m thousndth 0-3 centi c hundredth 0 - deci d tenth 0 - dek d ten 0 hecto h hundred 0 kilo k thousnd 0 3 meg M million 0 6 gig G billion 0 9 ter T trillion 0 pet P qudrillion 0 5 When you re given vrible with prefix you must convert it into its numericl equivlent in stndrd form before you use it in n eqution. FOLLOW THIS! lwys strt by replcing the prefix symbol with its equivlent multiplier. For exmple: 0.6 μ = 0.6 x km = 3 x 0 3 m 0 ns = 0 x 0-9 s DO NOT get tempted to follow this further (for exmple: 0.6 x 0-6 =.6 x 0-7 nd lso 0 x 0-9 s = 0-8 s) unless you re bsolutely confident tht you will do it correctly. It is lwys sfer to stop t the first step (0 x 0-9 s) nd type it like this into your clcultor. NOW TRY THIS!.4 kw = 0 μc = 4 cm = 340 MW = 46 pf = 0.03 m = 5 Gbytes = 43 kω = 0.03 MN = 0

11 Chpter 4: How to solve lengthy problem Problems in physics cn pper to be difficult t first sight. However, once you nlyse the problem in well-defined steps you should be ble to solve it without ny difficulty. The steps you need to follow re:. Identify the vribles you re given nd the ones you re sked to find. Convert ll units given to SI units 3. Give different symbol to ech vrible; try to stick to the well known symbols. To simplify, write the vlues for ech symbol (m=600) but don t worry bout writing the units t this stge. 4. Recognise which eqution/s to use. You do this by looking t wht vribles re vilble to you nd wht vribles you re sked to find. This is criticl stge; experience is the most importnt fctor here. This is why you need to prctise gin nd gin... This is lso why you need to KNOW LL YOUR EQUTIONS VERY WELL! 5. Find the logicl sequence for using these equtions in order to rech the desirble outcome. gin, experience is very importnt here! 6. Write the finl nswer nd dd the correct units. EXMPLE cr of mss 600 kg is trvelling t 0 ms -. When the brkes re pplied, it comes to rest in 0.0 km. Wht is the verge force exerted by the brkes? [3] (Note: there re mny wys to solve this problem nd in shorter method will be introduced.) STEP. mss= 600 kg initil velocity = 0 m s - finl velocity = 0 m s - stopping distnce = 0.0 km force =? STEP. re they ll in SI units? No. So... stopping distnce = 0.0 km = 0.0 x 0 3 m = 0 m STEP 3. m 600 u v s F?

12 STEP 4. E W (Chnge in energy = work done) W Fs (Work done = force x distnce in the direction of the force) EK mv (Kinetic energy = 0.5 x mss x velocity x velocity) STEP 5. Find the initil kinetic energy: E K mu Find the finl kinetic energy: E K mv Find the chnge in kinetic energy: E E K E K Equlise the chnge in kinetic energy with the work done nd rerrnge to find the force: E W E Fs F E s F F 3000 STEP 6. The finl nswer is F 3000N The negtive sign shows tht the force is in opposite direction to the velocity, i.e. it is the frictionl force tht stops the cr. NOW TRY THIS! girl diving from 5 m pltform wishes to know how fst she enters the wter. She is in the ir for.75 s nd dives from rest (with n initil speed of zero). Wht cn you tell her bout her entry speed? (g = 9.8 m s - ) [7] REMEMBER! Follow ll steps! Do not try to rush through! STEP.

13 STEP. STEP 3. STEP 4. STEP 5. STEP 6. 3

14 Chpter 5: How to recognise hidden mthemticl menings in sentences This chpter gives you first tste of the hidden mthemticl menings in sentences nd words. dd your own s you progress in the course. SENTENCE Chrge per unit time Metres per second per second Resistnce per unit length the product of voltge nd current The sum of powers of the lenses MTHEMTICL MENING Chrge divided by time Metres divided by second squred Resistnce divided by length Multiply voltge with current dd ll powers of lenses 4

15 Chpter 6: Logrithms 6. Definition of logrithms The use of logrithms sounds relly scry but once you lern the bsics you will be ble to mnipulte equtions with logrithms with no problem. This is the most importnt eqution tht summrises the ide behind ll logrithms: [4] x y log y x nd here re few exmples: 3 8 log 8 3 (you red: the logrithm of 8 to the bse is 3) log (red: the log of 000 to the bse 0 is 3) NOW TRY THIS! log log log log log log log Properties of logrithms ll the reltionships below re true, regrdless of the vlue of the bse. [5] log 0 log 5

16 log ( p q) log p log q log p q log p log q r log ( x ) rlog x 6.3 The two most fmous logrithms In physics the two most useful logrithms re: the Common logrithms which use bse 0 nd the Nturl logrithms which use the irrtionl number e (e=.78883) s the bse. To void complicting the equtions, the symbol for Common logrithms is log x nd the symbol for Nturl logrithms is ln x 0 log x insted of 6.4 How to solve problems with logrithms There re mny res of physics where you will encounter logrithms: sound levels, noise limittion on mximum bits per smple, rdioctive decy, dischrge of cpcitor etc. The importnt thing is to know wht to do in ll cses. When you re sked to mnipulte logrithmic eqution remember to USE the definition of logrithm s prt of the mnipultion process. For instnce, if you wnt to solve the eqution below with V totl s the subject you follow these steps: b log V V totl b totl b ( ) Vtotl Vnoise Vnoise V noise NOW TRY THIS!. Solve the sme eqution, but with V noise s the subject: 6

17 . The number of decibels d is given by the eqution d 0log 0 I I ) Rerrnge this eqution with I s the subject (the first step hs been done for you): I d d 0log0 log I 0 0 I I b) Now, rerrnge this eqution with I s the subject: 3. sound tht is on the threshold of udible intensity is 0 - W m - (like hering pin drop). This is tken s the bseline intensity I. pinful sound (like jet tking off) hs n intensity of bout 0 W m -. Wht is the sound level of the jet in db? [8]. The decibel formul is given in question bove. (ns: 30 db) 7

18 Chpter 7: Grphs 7. Types of grphs Grphs re VERY importnt in physics becuse they show ptterns between vribles. stright line grph tht strts from the (0,0) point is the best proof tht two vribles re directly proportionl. Stright line grph You should know from your mths tht the generl eqution for stright line is y x b cuts the y-xis., where is the grdient of the grph nd b is the point tht the line y b x You must lso know from your mths the eqution for hyperbol, prbol, n ellipse etc. Mke your own tble including ll the grph shpes you know nd their functions. Mke sure you include the logrithmic nd exponentil functions. Function Grph Exmple in physics y x b 8

19 7. How to choose the right grph for plotting lthough the bove list is importnt, when it comes to finding reltionship between two vribles the only grph tht cn show this very clerly is the stright line grph. EXMPLE Let s sy tht you wnt to prove the reltionship between the kinetic energy of n object nd its velocity. You plot velocity on the x-xis nd kinetic energy on the y-xis. You will get 9

20 curve which s you now is prbol (since the kinetic energy is directly proportionl to the squre of the velocity). Now let s sy you do nother experiment tht, unknown to you, lso follows the sme pttern. You will lso get curve when you plot the grph. Will you be ble to recognise tht this is prbol? Wht if it is curve tht is very close to prbol but not quite? Wht cn you do to be sure tht you hve crcked the reltionship? Think gin bout the exmple bove. If insted of plotting kinetic energy ginst velocity you plot kinetic energy ginst velocity squred wht will you get? You will get stright line through zero! Moreover, you will be certin tht the reltionship is tht: the kinetic energy is directly proportionl to the velocity squred. So wht hve we lerned so fr? LWYS IM T PLOTTING TWO VRIBLES THT WILL GIVE YOU STRIGHT LINE! Here re some exmples: To prove tht resistnce R is inversely proportionl to cross sectionl re, plot R ginst. This should give you stright line. To prove tht the squre of the period T of pendulum is directly proportionl to its length l plot either T ginst l or T ginst l NOW TRY THIS!. The pendulum eqution is: T l g ) Wht vribles should you plot ginst ech other in order to prove tht the period of the pendulum does not depend on its mss? Wht will the shpe of this grph be? b) Wht vribles should you plot ginst ech other to prove tht the period depends on the grvittionl field strength s shown by the eqution? 0

21 . The universl grvittionl lw is given by the eqution: F G mm r ) Wht vribles should you plot ginst ech other in order to prove tht the ttrctive force is directly proportionl to both msses of the objects? b) Wht vribles should you plot ginst ech other in order to prove tht the ttrctive force is inversely proportionl to the distnce squred between the objects? 7.3The significnce of the grdient During your course you will be sked to decide which grphs to plot in order to show reltionship or to clculte physicl constnt. We hve lredy noted how importnt it is to im t plotting grph tht will end up being stright line. This gives you definite nswer bout the reltionship between the two vribles. But there is more to it. The grdient of this line will give you informtion bout constnt in your experiment. EXMPLE Let s sy tht you wnt to mesure the grvittionl field strength of Erth with pendulum. T ginst l. The You vry the length nd mesure the period. You then decide to plot grph will be stright line. Wht will its grdient be? To find this, compre the pendulum eqution with the stright line eqution s shown below: T 4 l g y x b

22 I hope you cn see tht y corresponds to to zero, nd corresponds to 4 g grdient from your grph you will know the vlue of clculte g from this s: T, x corresponds to l, b corresponds. This tells you tht once you mesure the 4 g nd you will then be ble to grdient 4 4 g g grdient NOW TRY THIS! Try to find the grdient in ll the situtions listed below. The first three hve been done for you. Eqution Plot y ginst x grdient Constnt V x xis: current grdient R R R (for fixed resistor) I y xis: voltge V x xis: voltge R R grdient (for fixed resistor) I y xis: current R x xis: force l grdient y xis: extension E l E E grdient R L R L F x l mv Fs (stopping distncevelocity reltionship) xd L (double slit interference) xd L (double slit interference) x xis: L y xis: R x xis: y xis: R x xis: v y xis: s x xis: d y xis: x x xis: L y xis: x grdient (Young s modulus) (resistivity) grdient grdient F grdient grdient (resistivity) (fricition) (wvelength) (wvelength)

23 prt from its use s explined bove, the grdient in ll lines (curved or stright) corresponds to the derivtive of the function you plot. This is why if you plot time on the x- xis nd displcement on the y-xis the grdient corresponds to the velocity of the object. If the line is curved the grdient does not sty the sme, which mens tht it is equl to the instntneous velocity of the object. For the sme reson if you plot time on the x-xis nd velocity on the y-xis the grdient corresponds to the ccelertion of the object. If the line is curved the grdient does not sty the sme, which mens tht it is equl to the instntneous ccelertion of the object. 7.4The significnce of the re under grph The re between grph of y = f(x) nd the x-xis is equl to the definite integrl of the function. This formul gives positive result for grph bove the x-xis, nd negtive result for grph below the x-xis. [9] This is why the re under velocity-time grph is equl to the distnce covered by the object. If the grph is stright line then the re under cn be clculted very precisely s the re of tringle or trpezium etc. If the line is curve, the re is often estimted to good precision before it cn give you some useful informtion. You will use this ide more t level so for the moment we will leve it t this stge. 3