5. Pendulum Experimentln general, the period lor a pendulum formed by aweight suspended by a string of negligible mass is

z"^'[U\,where L is the length from t'fre pivot point tothe center of mass (actually, the center or percussion)of the weight, and g is the gravitational acceleration,about 9.8 m/sec2. Consequently, if the pendulum is 1

meter long, its period ,"itt oe z,r{J/g.a = 2'007.'., orabout 2 sec. This is the period for a complete back-and-forth swing. You must quadruple the length of apendulum to double its period. A pendulum hung fromthe ceiling will have a period slow enough to measurefairly preCisely. A good way to get more accuracy is tocouht the total time for 10 swings, then divide by 10'The period is roughly constant for any (moderate)ampiitude, as long as the amplitude is not too big. Thisfaci is not obvious to the uninitiated student, and isworth spending time showing. lt is quite dramatic towatch a pendulum take just as long to make 10 swingswith an amplitude of 2 cm as it does with amplitude 20or 30 cm.

6. Daylight Research ProjectThe following data were computed from actual sunriseand sunset times for San Antonio for each 10 days.You can get similar information for your locality fromthe local weather bureau, or newspaper office, or fromthe Nautical Almanac Office, U.S. Naval Observatory'Washington, DC 20390, or from the lnternet.

Day Mh Day Min DaY Min

7. Squeeze Theorem. Numericallya. Graph. t(x) = -2x2 + 8x - 2, g(x) = 2x2 + 2,

h(x) = {.x. Limits are all equal to 4.

Each function is continuous becausepolynomial function.f(x)<h(x)<g(x).

itisa

0 617 120 797 240 77210 623 130 811 250 75520 632 140 823 260 73830 645 150 833 270 720

160 840 280 703

c. x0.95 3.7950.96 3.83680.97 3.87820.98 3.91920.99 3.95981.00 41.01 4.03981.02 4.07921.03 4.11821.04 4.15681 .05 4.195

d. From the table, 6 = 0.01 or 0.02 willwork, but 0.03 is

too large.e. All the yalues of h(x) are between the corresponding

values of f(x) and g(x), and the three functions allapjrroach4asalimit.

8. Limit of (sin x)/x. Numericallya. The numbers are correct.b. x (sin x)/x

0.05 0.99958338541...0.04 0.99973335466...0.03 0.99985000674...o.o2 0.99993333466...0.01 0.99998333341...

Values are getting closer to 1, Q.E.D.c. Answers will vary according to calculator. For the Tl-

82 in rnalr mod6, starting i at 0 and using Ax = 10-7shows that all values round to 1 until x reaches 1.8 x104, which registers as 0.999999999999.

d. Answer will depend on calculator. For Tl-82 in taelemode, (sin 0.001)/0.001 is 0.999999833333, whichagrees exactly with the value published by NBS to12 places.

e. lf students have studied Taylor series (Chapter 12)before taking this course, they will be able to see thereason. The Taylor series for sin 0.001 is

0.001 -#.9-Y.= 0.00100 00000 00000 00000 000...

- 0.00000 0000'1 66666 66666 666...+ 0.00000 00000 00000 00833 333...

= 0.00099 99998 33333 34166 666...

h(x) g(x)3.8 3.8053.84 3.84323.88 3.88183.92 3.92083.96 3.9602444.04 4.04024.08 4.08084.12 4.12184.16 4.16324.2 4.205

40 660s0 67660 69370 711 190 83680 729 200 82890 747 210 816

170 842180 842

290 686300 669310 653320 639330 628

100 764 220 803 340 620110 780 230 789 350 615

360 615The graph shows a good fit to the data. But there is anoticeable deviation in the fall and winter, where theday is slightly longer than predicted. The main reasonforthe discrepancy, apparently, is the fact that in thefall and winter the earth is closer to the sun, and hencemoves slightly more rapidly through its angle with thesun than during the spring and summer.

hoblem Sei 3-8 Solutions Mqnuol 39