Solutions to: Unit Conversions and Significant Figures Problem Set S.E. Van Bramer (9/10/96)

Transcription

1 Solutions to: Unit Conversions and Significant Figures Problem Set S.E. Van Bramer (9/10/96) 1. Conversions: Express the following in the units asked for: a. Speed of light, 3.00 x 10 8 m s -1, in les per hour and meters/. One possible set of conversion factors is: m Double check with mathcad: 1in 2.4 cm 1ft 12 in ft 60 1 n 60 n = m = Remember, Mathcad does not work the significant figures. How many should there be in the answer? b. Speed of sound, 740 mph, in kilometers per hour and meters/. Convert using Ratio's: km km = km 10 3 m 1km 1 = m s c. Length of a C-C bond in diamond, (angstroms), in cm, m, and inches (watch significant figures on this one) A := m A m 1A = cm m A = m 1A The conversion factors are "exact" numbers, so they don't count for deterning the number of significant digits in the answer A m 1A Here, I set Mathcad 1in = to 9 in display extra 2.4 cm digits in the answer. How many significant figures should there be? How should you round this?

2 d. Temperature i. LN2, -19 C, in F and K LN2 C := 19 I am using the subscript C to indicate that this is the celcius temperature (Mathcad does not do temperature conversions) The formula to convert C to K is: K = C From the back of your book C = K By rearranging the above equation The Appropriate equation for this problem is: LN2 K := LN2 C LN2 K = 78.1 Temperature in kelvin The formula to convert C to F (From the back of your book) is: F = 9 C Apply to this problem: LN2 F := LN2 C + 32 LN2 F = 319 Temperature in Faenheit ii. Body Temp, 98 F, in C and K temp F := 98 The Conversion factor from the back of your book: C = 9 ( F 32) ( ) temp C := 9 temp F 32 temp C =

3 iii. Paper Flashpoint, 40 F, in C and K LN2 F := 40 LN2 C := ( 9 LN2 F 32) LN2 C = LN2 K := LN2 C LN2 K = iv. Derive Conversion for F to C based on freezing and boiling point of water. Freezing F := 32 Boiling F := 212 Freezing C := 0 Boiling C := 100 Each scale has a different number of degrees between freezing and boiling. This corresponds to different size "steps". Part of the conversion factor is to adjust the step scale. F steps := Boiling F Freezing F F steps = 180 C steps := Boiling C Freezing C C steps = 100 From this the conversion between step size (X F_C for F to C and X C_F for C to F) is: F steps = C steps X F_C C steps = F steps X C_F These rearrange to: F steps C steps X F_C := X C_F := C steps F steps X F_C = 1.8 X C_F = 0.6 X F_C = 9 1 X C_F = 1 9 Finally we need to account for the different starting points for the two scales When converting F to C, first subtract 32, so both scales are at 0 When converting C to F, add 32 when you are finished. 2. What is the volume of object that is 83.6 cm long, cm wide, and 1.2 inches thick? What is the volume after.74 cm is cut off from the long edge? length := 83.6 cm width := cm height := 1.2 in volume := length width height After converting all tee values to the same units volume cm 3 = How many significant figures should this number have? If you cut.74 cm from the long edge, the new length is: length := length.74 cm volume := length width height Which gives a new volume of: volume = cm 3 How many significant digits should this number have?

4 3. Solve the following problems and express the results with the correct number of significant figures = = = = = = ( ) ( ) = The mass of the sun is 1.989x10 30 kg and the radius is km. More on the sun. What is the average density of the sun in the following units: a. kg/km3 = b.kg/m3 = c. kg/cm3 = d. g/cm3 = mass := kg mass = gm The equation for the volume of a sphere is: 4 r := km volume := 3 π r3 Convert units: to cubic meters: km 3 = 1kmkm km km 3 = m volume = km 3 ( ) ( 10 3 m) ( 10 3 m) km 3 = m km m 3 km 3 = m 3 volume = m 3 to cubic centimeters: m 3 = ( ) ( 10 2 cm) ( 10 2 cm) cm m 3 = cm m cm 3 m 3 = cm 3 volume = cm 3

5 So the density of the sun is: mass density := volume density = kg km 3 How does this density relate to values you are faliar density = kg m -3 with? How many significant digits should the solution density = kg cm 3 have? density = gm cm 3. In the opening scenes of the movie Raiders of the Lost Ark, Indiana Jones tries to remove a gold idol from a booby-trapped pedestal. He replaces the idol with a bag of sand of approximately equal volume. (Density of gold = g/cm3; density of sand 3 g/cm3.) {from Zumdahl, Chestry, Saunders (1994). a. Did he have a reasonable chance of not activating the mass-sensitive booby trap? NO, the gold is 7 times more dense than the sand. The Idol would be much heavier. b. in a later scene he and an unscrupulous guide play catch with the idol. Assume that the volume of the idol is about 1.0 L. If it were solid gold, what mass would the idol have? Is playing catch with it plausible? volume := 1 liter volume = cm 3 density := gm cm 3 mass density = volume mass := density volume mass = kg mass = lb Can you toss a 40 pound bag of cement?

6 6. The world record for the hundred meter dash is 9.79 s (mens) and s (womens). a. What is the average speed in m/s b. What is the average speed in km/h c. What is the average speed in ft/s d. What is the average speed in /h? e. At this speed, how long would it take to run 1.00 x 10 2 yards? Men's Record := 100 m 9.79 speed := time m s km ft You can perform the unit conversions here. How many significant figures should the answer have? Be careful, how "precise" and accurate is the 100 in the "100 meter". Think about where the uncertainty is. Women's Record For 100 yards, convert to metric: speed := 100 yd = m time = s := 100 m speed := time m s km For 100 yards, convert to metric: speed ft := 100 yd = m time = s

7 7. The mercury concentration in a polluted lake is µg Hg/L, what is the total mass in kilograms of mercury in the lake if the lake is approximately circular with a diameter of 10 les and an average depth of 0 ft? First, find the surface area of the lake: diameter := 10 = m diameter = m diameter = m r := 1 2 diameter r = m area := π r 2 area = m 2 From the surface area and the depth, find the volume depth := 0 ft depth = 0 ft depth = 1.24 m volume := 12 in 1ft area depth 2.4 cm 1in volume = m 3 Next convert the concentration to cubic meters: µg := 10 6 gm concentration Hg := µg liter concentration Hg = µg liter 10 6 gm µg concentration Hg = gm m 3 1 liter 1000 ml 1mL 1cm 3 The concentration is the mass of mercury in each cubic meter of water. To find the total mass: mass Hg := concentration Hg volume mass Hg = gm m 3 mass Hg = kg mass Hg = ton This is a rather large amount of mercury. How many significant figures should this answer have?