Physics 161: Problem Set 2 - SOLUTIONS

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1 Phyic 161: Problem Set 2 - SOLUTIONS April 7, (1 pt. each). Scientific notation and calculation (a) ( ) ( ) = (b) ( ) ( ) = = 0.02 (c) ( ) ( ) (0.3) = (3pt.). kwh/day i a unit of power. We know thi becaue the kw i a unit of power, and h/day i unitle (h/day ha unit of time/time). Thu, the dimenion of kwh/day are the ame a the dimenion of kw, which again, i a unit of power. Another way to think about thi: kw i a unit of power, o kwh i a unit of power time = energy. kwh / day i therefore a unit of energy/time, i.e. power. 3 (2pt.). There are everal way to olve thi problem, all of which involve the fundamental relationhip Power = Energy / Time. (1) Let expre our 30 W a 30 J/ (ince a Watt i a J/). We can convert thi to J/ 30J 60 = J. The only way J and 90, 000J can be multiplied to get J unit of time i: 90, 000J = 50 ute (2) Another approach: Power = Energy / Time, o Power Time = Energy, and Time = Energy / Power. Therefore Time = 90,000 J / 30 W = 90,000 J / 30 J/ = 3,000 econd. 3,000 econd 1 / 60 ec = 50 ute. 4. Common device (8pt.) Again, we make ue of Power = Energy / Time, i.e. Power Time = Energy. Note that to calculate the energy, your time mut be in econd, and power in Watt. power (W) runtime runtime (econd) energy (J) laptop 65 8 hr tv 74 2 hr microwave digital clock 8 24 hr

2 5. Horepower (a, 1 pt.) 300hp 750W hp = W. Note that thi i the only way to ue the converion factor for which the unit work out. (b, 4 pt.) Again, we make ue of Power = Energy / Time. Note alo that 1 W = 1 J/. Engine: E = J = J. To do thi without a calculator: , and = Then E = J = J Fridge: E = 500 J 60 24hr hr 60ec = J. Without a calculator: 24hr hr, 60 hr 60ec = hr. Then E = J hr hr = J So, the engine running for 40 ute ue an order of magnitude more energy than the fridge running all day! 6 (8 pt.). Energy denity of variou food. Mot nutrition fact heet will give the erving ize in gram (g), and the energy content in Calorie. To find the energy denity in J/kg, you convert the Calorie to Joule with the converion factor 4184J/Cal, then convert gram to kg with the converion factor.001kg/g, then take the ratio of thee two number. Calorie erving ize (g) energy denity (J/kg) banana Big Mac French Fry Skittle alad (5pt.) Converting energy unit. (a, 1pt.) Btu J Btu = J (b, 4pt.) Note that we know the power in Joule / year. We want to convert thi to Watt - i.e. Joule per econd. So: J yr 1yr 365d 1d 24hr 1hr 3600 = J = 11400W = 11.4kW 8 (8 pt.) Energy uage in Oregon. (a, 4pt.). Source: Energy Information Aditration, State Energy Profile, Oregon: http : //tonto.eia.doe.gov/tate/tate energy prof ile.cf m?id = OR 2

3 Total energy conumption per capita in the year 2005: 302millionBtu/yr = Btu yr J yr 1yr 365d 1hr 3600 = 10100W = 10.1kW The average Oregonian conume lightly le power (about 10% le) than the average U.S. Citizen. (b, 4pt.). Energy Conumed by Source in the Year Source: Energy Information Aditration: State Data for Conumption and Sale, Table S1:Energy Conumption Etimate by Source and End-Ue Sector, 2005 http : // um/plain html/um btu 1.html To find the percentage contributed by each ource, we divide the total energy conumed from that ource (the value given in column 2) by the total energy conumed, which i found in the ame data table to be 1, 095.7trillionBtu = 1, Btu, then multiply by 100 to convert the fraction to a percentage. The fourth and fifth column give the correponding US value for comparion (or you can compare to the value given in cla or Wolfon); the mot triking difference that hould be apparent i that Oregon ue much more hydroelectric power and much le coal than the US average. Oregon alo ue no nuclear power, while nuclear make up almot 8% of the US total. Finally, Oregon ue almot twice a much bioma derived power, although thi ource i till a mall percentage (le than 5%) of the total. Source Energy Conumed, Percent of Energy Conumed, % of total, OR (trillion Btu) Total, OR US(trillion Btu) US All ource 1, % % combined Coal % 22, % Natural Ga % 22, % Petroleum % 40, % Nuclear Elec % 8, % Power Hydroelectric % 2, % Bioma % 2, % Other % % 9 (4 pt. total). Some math quetion (a) We know that Z = XY 4. In the problem we know that for ome value of X, Z = 32 and Y = 2. Now Y i changed to 4, and X remain the ame. How can we find the new value of Z? The traight forward way to do thi i to firt olve for X. Then when Y i changed, we can olve for 3

4 Z uing the unchanged value of X Then when Y i changed to 4: X = Z Y 4 = = = 2 Z = = = 512 (b) There an eaier way to olve thi ort of problem than the brute-force method. Z = XY 4, i.e. Z = X Y Y Y Y. If Y become twice a big, we can imagine replacing each Y with 2Y, writing X 2Y 2Y 2Y 2Y. Thi give u an extra factor of compared to what we had before, o Z i 2 4 time bigger (i.e. 16). Note that by thi logic, if Y were 3 time bigger, Z would be = 3 4 = 81 time bigger. Another, more algebraic explanation: Y. We know they atify the equation Thi can be re-written a Suppoe that we are given the initial value for Z and Z = XY 4 X = Z Y 4 (1) Now, aume the value of Y i changed to ome new number which I denote Y, and X remain unchanged. In thi cae Z will take on ome new value which I ll denote Z. Thi give the equation Z = X(Y ) 4 which again can be re-written a X = Z (Y ) 4 (2) Notice that equation (2) and equation (1) have the ame value, namely X, on the left hand ide. Thi tell u that the right hand ide of equation (2) mut be equal to the right hand ide of equation (1): Z Y 4 = Z (Y ) 4 Solving for Z (the thing we want to calculate): ( ) Y Z 4 = Z (3) Y 4

5 Since the problem give you the value for Z, Y, and Y, Z can be calculated immediately with no intermediate tep. In all three of the example you re aked to compute here, Y = 2Y. Plugging thi into the equation give Z = Z ( 2Y Y ) 4 = Z ( 2 4) = 16Z Uing thi, all of the computation are traightforward: Y = 1, Z = 3, Y = 2 Z = 16Z = 46 Y = 5, Z = 1250, Y = 10 Z = 16Z = Y = 3, Z = 257, Y = 6 Z = 16Z =