Putting Scientific Notation to Work

Transcription

1 10 Putting Scientific Ntatin t Wrk Physics deals with sme very large and very small numbers. T wrk with such numbers, yu use scientific ntatin. Scientific ntatin is expressed as a number multiplied by a pwer f 10. Fr example, suppse yu re measuring the mass f an electrn in the MKS system. Yu put an electrn n a scale (in practice, electrns are t small t measure n a scale yu have t see hw they react t the pull f magnetic r electrstatic frces in rder t measure their mass) and yu measure the fllwing: kg What the heck is that? That s a lt f zers, and it makes this number very unwieldy t wrk with. Frtunately, yu knw all abut scientific ntatin, s yu can cnvert the number int the fllwing: kg That is, 9.1 multiplied by a pwer f 10, Scientific ntatin wrks by extracting the pwer f 10 and putting it n the side, where it s handy. Yu cnvert a number t scientific ntatin by cunting the number f places yu have t mve the decimal pint t get the first digit in frnt f that decimal pint. Fr example, is because yu mve the decimal pint tw places t the right t get 5.0. Similarly, 500 is because yu mve the decimal pint tw places t the left t get 5.0. Check ut this practice questin abut scientific ntatin: Q. What is in scientific ntatin? A. The crrect answer is Yu have t mve the decimal pint five times t the right t get 3.7.

2 Chapter 1: Getting Started with Physics What is in scientific ntatin? 6. What is in scientific ntatin? 7. What is in scientific ntatin? 8. What is in scientific ntatin?

3 12 Cnverting between Units Physics prblems frequently ask yu t cnvert between different units f measurement. Fr example, yu may measure the number f feet yur ty car ges in three minutes and thus be able t calculate the speed f the car in feet per minute, but that s nt a standard unit f measure, s yu need t cnvert feet per minute t miles per hur, r meters per secnd, r whatever the physics prblem asks fr. Fr anther example, suppse yu have 180 secnds hw much is that in minutes? Yu knw that there are 60 secnds in a minute, s 180 secnds equals three minutes. Here are sme cmmn cnversins between units: 1 m = 100 cm = 1000 mm (millimeters) 1 km (kilmeter) = 1000 m 1 kg (kilgram) = 1000 g (grams) 1 N (Newtn) = 10 5 dynes 1 J (Jule) = 10 7 ergs 1 P (Pascal) = 10 ba 1 A (Amp) =.1 Bi 1 T (Tesla) = 10 4 G (Gauss) 1 C (Culmb) = Fr The cnversin between CGS and MKS is almst always just a factr f 10, s cnverting between the tw is simple. But what abut cnverting t and frm the FPI system? Here are sme handy cnversins that yu can cme back t as needed: Length: 1 m = 100 cm 1 km = 1000 m 1 in (inch) = 2.54 cm 1 m = in 1 mile = 5280 ft = km 1 Å (angstrm) = m Mass: 1 kg = 1000 g 1 slug = kg 1 u (atmic mass unit) = kg Frce: 1 lb (pund) = N 1 N = 10 5 dynes 1 N = lb Energy: 1 J = 10 7 ergs 1 J = ft-lb

4 Chapter 1: Getting Started with Physics 13 1 BTU (British Thermal Unit) = 1055 J 1 kwh (kilwatt hur)= J 1 ev (electrn Vlt) = J Pwer: 1 hp (hrsepwer) = 550 ft-lb/s 1 W (Watt)= ft-lb/s Because cnversins are such an imprtant part f physics prblems, and because yu have t keep track f them s carefully, there s a systematic way f handling cnversins: Yu multiply by a cnversin cnstant that equals ne, and where the units yu dn t want cancel ut. Q. A ball drps 5 meters. Hw many centimeters did it drp? A. The crrect answer is 500 centimeters. T perfrm the cnversin, yu d the fllwing calculatin: 50. meters# 100 centimeters = 500 centimeters meters Nte that 100 centimeters divided by 1 meter equals 1 because there are 100 centimeters in a meter. In the calculatin, the units yu dn t want meters cancel ut. 9. Hw many centimeters are in 2.35 meters? 10. Hw many secnds are in 1.25 minutes?

5 Hw many inches are in 2.0 meters? 12. Hw many grams are in 3.25 kg? Cnverting Distances Smetimes yu have t make multiple cnversins t get what yu want. That demands multiple cnversin factrs. Fr example, if yu want t cnvert frm inches t meters, yu can use the cnversin that 2.54 centimeters equals 1 inch but then yu have t cnvert frm centimeters t meters, which means using anther cnversin factr. Try yur hand at this example questin that invlves multiple cnversins: Q. Cnvert 10 inches int meters. A. The crrect answer is m. 1. Yu knw that 1 inch = 2.54 centimeters, s start with that cnversin factr and cnvert 10 inches int centimeters: 10 in # 254. # cm = cm 1 in 2. Cnvert 25.4 cm int meters by using a secnd cnversin factr: 10 in # 254. cm # 1 m = m 1 in 100 cm

6 Chapter 1: Getting Started with Physics Given that there are 2.54 centimeters in 1 inch, hw many centimeters are there in 1 yard? 14. Hw many centimeters are in a kilmeter? 15. Hw many inches are in an angstrm, given that 1 angstrm (Å) = 10 8 cm? 16. Hw many inches are in a meter, given that there are 2.54 cm in 1 inch?

7 16 Cnverting Times Physics prblems frequently ask yu t cnvert between different units f time: secnds, minutes, hurs, and even years. These times invlve all kinds f calculatins because measurements in physics bks are usually in secnds, but can frequently be in hurs. Q. An SUV is traveling kilmeters per secnd. What s that in kilmeters per hur? A. The crrect answer is 100 km/hr. 1. Yu knw that there are 60 minutes in an hur, s start by cnverting frm kilmeters per secnd t kilmeters per minute: - 2. km 60 sec 278# 10 # = 166. km/ minute sec 1 minute 2. Because there are 60 minutes in an hur, cnvert this t kilmeters per hur using a secnd cnversin factr: - 2. km 60 sec 60 minutes 278# 10 # # = 100 km/ hr sec 1 minute 1 hur 17. Hw many hurs are in 1 week? 18. Hw many hurs are in 1 year?

8 Cunting Significant Figures Chapter 1: Getting Started with Physics Yu may plug numbers int yur calculatr and cme up with an answer like , but that number isn t likely t please yur instructr. Why? Because in physics prblems, yu use significant digits t express yur answers. Significant digits represent the accuracy with which yu knw yur values. 17 Fr example, if yu knw nly the values yu re wrking with t tw significant digits, yur answer shuld be 1.5, which has tw significant digits, nt , which has 13! Here s hw it wrks: Suppse yu re tld that a skater traveled 10.0 meters in 7.0 secnds. Nte the number f digits: The first value has three significant figures, the ther nly tw. The rule is that when yu multiply r divide numbers, the result has the number f significant digits that equals the smallest number f significant digits in any f the riginal numbers. S if yu want t figure ut hw fast the skater was ging, yu divide 10.0 by 7.0, and the result shuld have nly tw significant digits 1.4 meters per secnd. Zers used just t fill ut values dwn t (r up t) the decimal pint aren t cnsidered significant. Fr example, the number 3600 has nly tw significant digits by default. That s nt true if the value was actually measured t be 3600, f curse, in which case it s usually expressed as 3600.; the final decimal indicates that all the digits are significant. On the ther hand, when yu re adding r subtracting numbers, the rule is that the last significant digit in the result crrespnds t the right-mst clumn in the additin r subtractin. Hw des that wrk? Take a lk at this additin example: S is the result 24.83? N, it s nt. The 12 has n significant digits t the right f the decimal pint, s the answer shuldn t have any either. That means yu shuld rund the value f the result up t 25. Runding numbers in physics wrks as it usually des in math: When yu want t rund t three places, fr example, and the number in the furth place is a five r greater, yu add ne t the third place (and ignre r replace with zers any fllwing digits). Q. Yu re multiplying by 9.7. What shuld yur answer be, keeping in mind that yu shuld express it in significant digits? A. The crrect answer is The calculatr says that the prduct is Yur result has t have the same number f significant digits as the least number f any tw values yu multiplied. That s tw here (because f 9.7), s yur answer runds up t 120.

9 What is 19.3 multiplied by 26.12, taking int accunt significant digits? 20. What is the sum f 7.9, 19, and 5.654, taking int accunt significant digits? Cming Prepared with Sme Algebra It s a fact f life: Yu need t be able t d algebra t handle physics prblems. Take the fllwing equatin, fr example, which relates the distance smething has traveled (s) t its acceleratin and the time it has been accelerated: 1 2 s= at 2 Nw suppse that the physics prblem actually asks yu fr the acceleratin, nt the distance. Yu have t rearrange things a little here t slve fr the acceleratin. S when yu multiply bth sides by 2 and divide bth sides by t 2, here s what yu get: 2 s 2 2 = 2 t t 1 2 $ $ $ $ a t Cancelling ut and swapping sides, yu slve fr a like this: a 2 s t = $2 2

10 Chapter 1: Getting Started with Physics 19 S that s putting a little algebra t wrk. All yu had t d was mve variables arund the equatin t get what yu want. The same apprach wrks when slving physics prblems (mst f the time). On the ther hand, what if yu had t slve the same prblem fr the time, t? Yu wuld d that by rearranging the variables like s: t= 2s/ a The lessn in this example is that yu can extract all three variables distance, acceleratin, and time frm the riginal equatin. Shuld yu memrize all three versins f this equatin? Of curse nt. Yu can just memrize the first versin and use a little algebra t get the rest. The fllwing practice questins call n yur algebra skills: Q. The equatin fr final speed, v f, where the initial speed was v, the acceleratin was a, and the time was t is v f v = at. Slve fr acceleratin. A. The crrect answer is a= _ v f-vi/ t T slve fr a, divide bth sides f the equatin by time, t. 21. The equatin fr ptential energy, PE, f a mass m at height h, where the acceleratin due t gravity is g, is PE = m g h. Slve fr h. 22. The equatin relating final speed, v f, t riginal speed, v, in terms f acceleratin a and distance s is v f 2 v 2 = 2as. Slve fr s.

11 The equatin relating distance s t acceleratin a, time t, and speed v is s= v$ t+ 1 2 a t 2. Slve fr v. 24. The equatin fr kinetic energy is 1 2 KE 2 m v = $ $. Slve fr v, given KE and m. Being Prepared with Trignmetry Physics prblems als require yu t have sme trignmetry under yur belt. T see what kind f trig yu need, take a lk at Figure 1-1, which shws a right triangle. The lng side is called the hyptenuse, and the angle between x and y is 90. h y Figure 1-1: A triangle. θ x

12 Chapter 1: Getting Started with Physics 21 Physics prblems require yu t be able t wrk with sines, csines, and tangents. Here s what they lk like fr Figure 1-1: sin θ = y/h cs θ = x/h tan θ = y/x Yu can find the length f ne side f the triangle if yu re given anther side and an angle (nt including the right angle). Here s hw t relate the sides: x = h cs θ = y/tan θ y = h sin θ = x tan θ h = y/sin θ = h/cs θ And here s ne mre equatin, the Pythagrean Therem. It gives yu the length f the hyptenuse when yu plug in the ther tw sides: 2 2 h= x + y 25. Given the hyptenuse h and the angle θ, what is the length x equal t? 26. If x = 3 and y = 4, what is the length f h?

13 22 Answers t Prblems abut Getting Started with Physics The fllwing are the answers t the practice questins presented earlier in this chapter. Yu see hw t wrk ut each answer, step by step. a grams The unit f mass in the CGS system is the gram. b secnds The unit f time in the MKS system is the secnd. c Newtns The unit f frce in the MKS system is the Newtn. d ergs The unit f energy in the CGS system is the erg. e Yu have t mve the decimal pint three places t the right. f Yu have t mve the decimal pint five places t the left. g Yu have t mve the decimal pint seven places t the right. h Yu have t mve the decimal pint three places t the left. i 235 cm Cnvert 2.35 meters int centimeters: 235. m# 100 cm = 235 cm 1 m j 75 sec Cnvert 1.25 minutes int secnds: 125. min # 60 sec = 75 sec 1 min k 78.6 in Cnvert 2.0 meters int inches: 20. m # in = in 1 m l 3250 g Cnvert 3.25 kilgrams int grams: 1000 g 325. kg# = 3250 g 1 kg

14 Chapter 1: Getting Started with Physics 23 m 91.4 cm 1. 1 yard is 3 feet, s cnvert that t inches: 3 ft # 12 in = 36 in 1 ft 2. Use a secnd cnversin factr t cnvert that int centimeters: n km 1. Cnvert 1 centimeter t meters: 12 in 3 ft # # 254. cm = cm 1 ft 1 in 1 cm 1 m - # = 10. # 10 2 m 100 cm 2. Use a secnd cnversin factr t cnvert that int kilmeters: in 1. Cnvert 1 angstrm t centimeters: 1 m 1 cm 1 km - # # = 10. # 10 5 km 100 cm 1000 m A # 10 cm = 10 1 A 2. Use a secnd cnversin factr t cnvert that int inches: p 39.3 in 1. Cnvert 1 meter int centimeters: - 8 cm cm 1 A # # 10. in = 40. # cm 1 A 1 m # 100 cm = 100 cm 1 m 2. Use a secnd cnversin factr t cnvert that int inches: q 168 hurs 1. Cnvert 1 week int days: 100 cm 1 m # # 1 in = in 1 m 254. cm 7 days 1 week # = 7 days 1 week 2. Use a secnd cnversin factr t cnvert that int hurs: r 8760 hurs 1. Cnvert 1 year int days: days 1 week # # 24 hurs = 168 hurs 1 week 1 day 365 days 1 year # = 365 days 1 year 2. Use a secnd cnversin factr t cnvert that int hurs: 365 days 1 year # # 24 hurs = 8760 hurs 1 year 1 day in

15 24 s The calculatr says the prduct is has three significant digits, and has fur, s yu use three significant digits in yur answer. That makes the answer 504. t Here s hw yu d the sum: The value 19 has n significant digits after the decimal place, s the answer shuldn t either, making it 33 ( runded up). u h = PE/mg Divide bth sides by mg t get yur answer. v w 2 2 vf - v 2 a = s Divide bth sides by 2a t get yur answer. s t - 1 at = v 2 1. Subtract at 2 /2 frm bth sides: 2. Divide bth sides by t t get yur answer. 1 2 s- at = vt 2 x v = 2 m KE 1. Multiply bth sides by 2/m: 2. Take the square rt t get yur answer. 2 2 m KE = v y x = h cs θ Yur answer cmes frm the definitin f csine. A 5 1. Start with the Pythagrean therem: 2 2 h= x + y 2. Plug in the numbers, and wrk ut the answer: 2 2 h= = 5

16 Chapter 2 The Big Three: Acceleratin, Distance, and Time In This Chapter Thinking abut displacement Checking ut speed Remembering acceleratin Being able t cnnect displacement, speed, and acceleratin is fundamental t wrking with physics. These things cncern peple every day, and physics has made an rganized study f them. Prblems that cnnect displacement, speed, and acceleratin are all abut understanding mvement, and that s the tpic f this chapter putting numbers int the discussin. Yu ll ften find physics prblems abut cars starting and stpping, hrses racing, and rcket ships zming back and frth. And after yu finish this chapter, yu ll be a real pr at slving them. Frm Pint A t B: Displacement Displacement ccurs when smething mves frm here t there. Fr example, suppse that yu have a ball at the zer psitin, as in Figure 2-1A. A meters Figure 2-1: A mving ball. B meters Nw suppse that the ball rlls ver t a new pint, 3 meters t the right, as yu see in Figure 2-1B. The ball is at a new lcatin, s there s been displacement. In this case, the displacement is just 3 meters t the right. In physics terms, yu ll ften see displacement referred t as the variable s. In this case, s = +3 meters.

17 26 Like any ther measurement in physics, displacement is always expressed in units, usually centimeters r meters, as in this example. Of curse, yu als can use kilmeters, inches, feet, miles, r even light years (the distance light travels in ne year 5,865,696,000,000 miles). The fllwing example questin fcuses n displacement. Q. Yu ve taken the pineers advice t G West. Yu started in New Yrk City and went west 10 miles the first day, 14 miles the next day, and then back east 9 miles n the third day. What is yur displacement frm New Yrk City after three days? A. s = 15 miles west f New Yrk City 1. Yu first went west 10 miles, s at the end f the first day, yur displacement was 10 miles west. 2. Next, yu went west 14 days, putting yur displacement at miles = 24 miles west f New Yrk City. 3. Finally, yu traveled 9 miles east, leaving yu at 24 9 = 15 miles west f New Yrk City. S s = 15 miles west f New Yrk City. 1. Suppse that the ball in Figure 2-1 nw mves 1 meter t the right. What is its new displacement frm the rigin, 0? 2. Suppse that the ball in Figure 2-1, which started 4 meters t the right f the rigin, mves 6 meters t the left. What is its new displacement frm the rigin in inches?

18 Reading That Speedmeter Chapter 2: The Big Three: Acceleratin, Distance, and Time In physics terms, what is speed? It s the same as the cnventinal idea f speed: Speed is displacement divided by time. 27 Fr example, if yu went a displacement s in a time t, then yur speed, v, is determined as fllws: v = s t Technically speaking, speed is the change in psitin divided by the change in time, s yu als can represent it like this if, fr example, yu re mving alng the x axis: v x xf- x = = t t - t f Q. Suppse that yu want t drive frm New Yrk City t Ls Angeles t visit yur uncle s family, a distance f abut 2781 miles. The trip takes yu fur days. What was yur speed in miles per hur? A. v x f = = =. t x - x t - t f miles per hur 1. Start by figuring ut yur speed (the distance traveled divided by the time taken t travel that distance): 2781 miles = days 2. Okay, the speed is , but what? This slutin divides miles by days, s it s miles per day nt exactly a standard unit f measurement. S what is that in miles per hur? T determine that, yu cancel days ut f this equatin and put in hurs. Because 24 hurs are in a day, yu can multiply as fllws (nte that days cancel ut, leaving miles ver hurs, r miles per hur): 2781 miles 1 day # = miles per hur 4 days 24 hurs S yur speed was miles per hur. That s yur average speed, averaged ver bth day and night.

19 28 3. Suppse that yu used yur new SpeedPass t get yu thrugh the tllbths at bth ends f yur trip, which was 90 miles n the turnpike and tk yu 1 hur and 15 minutes. On yur return hme, yu re surprised t find a traffic ticket fr speeding in the mail. Hw fast did yu g, n average, between the tllbths? Was the turnpike authrity justified in sending yu a ticket, given that the speed limit was 65 mph? 4. Suppse that yu and a friend are determined t find ut whse car is faster. Yu bth start yur trips in Chicag. Driving nnstp, yu reach Ls Angeles a distance f 2018 miles in 1.29 days, and yur friend, als driving nnstp, reaches Miami a distance f 1380 miles in 0.89 days. Whse car was faster? Putting Pedal t Metal: Acceleratin In physics terms, acceleratin is the amunt by which yur speed changes in a given amunt f time. In terms f equatins, it wrks like this: a = v t Given initial and final velcities, v and v f, and initial and final times ver which yur speed changed, t and t f, yu can als write the equatin like this: a v v f- v = = t t - t T get the units f acceleratin, yu divide speed by time as fllws: v f- v a = t - t f f distance/ time = = distance 2 time time Distance ver time squared? Dn t let that thrw yu. Yu end up with time squared in the denminatr just because it s velcity divided by time that s smething yu get used t when slving physics prblems. In ther wrds, acceleratin is the rate at which yur speed changes because rates have time in the denminatr. S fr acceleratin, yu can expect t see units f meters per secnd 2, r centimeters per secnd 2, r miles per secnd 2, r feet per secnd 2, r even kilmeters per hur 2.

20 Chapter 2: The Big Three: Acceleratin, Distance, and Time 29 Q. Suppse that yu re driving at 75 miles an hur and suddenly see red flashing lights in the rearview mirrr. Great, yu think, and yu pull ver, taking 20 secnds t cme t a stp. Yu culd calculate hw quickly yu decelerated as yu were pulled ver (infrmatin abut yur law-abiding tendencies that, n dubt, wuld impress the fficer). S just hw fast did yu decelerate, in cm/sec 2? A. / sec a v 3350 cm = = = 168 cm/ sec t 20 secnds 1. First cnvert t miles per secnd: 2 75 miles 1 hur 1 minute - # # = = # 10 2 miles per secnd hur 60 minutes 60 secnds 2. Cnvert frm miles per secnd t inches per secnd: # 10 miles 5280 feet # # 12 inches = 1318 in secnd 1 mile 1 ft secnd 3. Yur speed was 1318 inches per secnd. What s that in centimeters per secnd? # 10 miles 5280 feet 12 inches # # # centimeters = 3350 cm secnd 1 mile 1 ft inch secnd 4. What was yur acceleratin? That calculatin lks like this: a v v - v = = t t - t f cm/ secnd = - =-168 cm/ sec 20 secnds f 2 In ther wrds, 168 cm/sec 2, nt +168 cm/sec 2. There s a big difference between psitive and negative in terms f slving physics prblems and in terms f law enfrcement. If yu accelerated at +168 cm/sec 2 instead f accelerating at 168 cm/sec 2, yu d end up ging 150 miles per hur at the end f 20 secnds, nt 0 miles per hur. In ther wrds, the sign f the acceleratin tells yu hw the speed is changing. A psitive acceleratin means that the speed is increasing in the psitive directin, and a negative acceleratin (als knwn as deceleratin) tells yu that the speed is increasing in the negative directin.

21 30 5. A rcket ship is ging t land n the mn in exactly 2 hurs. There s nly ne prblem: It s ging 17,000 miles an hur. What des its deceleratin need t be, in miles per secnd 2, in rder t land n the mn safely at 0 miles per hur? 6. Yu re stpped at a red light when yu see a mnster SUV careening tward yu. In a lightning calculatin, yu determine yu have 0.8 secnds befre it hits yu and that yu must be ging at least 1.0 miles an hur frward at that time t avid the SUV. What must yur acceleratin be, in miles per hur 2? Can yu avid the SUV? 7. A bullet cmes t rest in a blck f wd in secnds, with an acceleratin f meters per secnd 2. What was its riginal speed, in meters per secnd? 8. The light turns red, and yu cme t a screeching halt. Checking yur stpwatch, yu see that yu stpped in 4.5 secnds. Yur deceleratin was miles per secnd 2. What was yur riginal speed in miles per hur?