Dimensional Analysis Part IV. Combination Units Dimensional Analysis IV. Combination Units Typical problem: The speed of a car is 60 miles/hour; express this speed in km/min. The speed of a car is 60 miles/hour; express this speed in km/min. Steps to solve 1. Identify starting units 2. Identify ending units 3. Identify unit conversion factors for both the numerator and denominator of the unit (you may need more than one) 4. Arrange conversion factors in an equation so they cancel out all the starting units and changes them into final units (No numbers) 5. Put in the numbers and solve the equation The speed of a car is 60 miles/hour; express this speed in km/min. Step 1. Identify starting units miles/hour The speed of a car is 60 miles/hour; express this speed in km/min. Step 2. Identify ending units km/min The speed of a car is 60 miles/hour; express this speed in km/min. 3. Identify unit conversion factors for both the numerator and denominator of the unit (you may need more than one) End Start kilometers/minute Miles/hour Numerator Miles to kilometers 1 kilometer = 0.62137 mile Denominator hours to minutes 1 hour = 60 minutes The speed of a car is 60 miles/hour; express this speed in km/min. 4. Arrange conversion factors in an equation so they cancel out all the starting units and changes them into final units (No numbers) miles/hour x (km/mile) x (hour/minute) Work on numerator and denominator separately. Notice how the unit to cancel the denominator has to be on the upper part of the conversion fraction The speed of a car is 60 miles/hour; express this speed in km/min. Step 5: Put in actual numbers and solve the problem 60 miles/hr x (1 km/0.62137 mi) x (1 hr/60 min) = 1.609347 km/minute =2 km/min (rounding to 1 significant figure) Practice Problems The next three pages have practice problems, try out the problems and see if you are doing them right The speed of light is 2.998x108 m/s. What is this in miles/hr Step 1: Starting Units meters and seconds Step 2: Ending Unit miles and hours Step 3: Unit conversions 1 km = 0.62137 mi; 1 km = 1x103 m 60 seconds = 1 minute; 60 minutes = 1 hour Step 4: Equation m/s x (km/m) x (mi/km) x (s/min) x (min/hr) Step 5: Solve 2.998x108 m/s x ( 1km/1x103m) x (0.62137 mi/1 km) x (60 sec/minute) x (60 minutes/hour) = 6.706322136x108 miles/hour 6.706x108 miles/hour (rounded to 4 sig fig) If I swim 2,100 meters/day, what is this in miles/week Step 1: Starting Units m/day Step 2: Ending Units miles/week Step 3: Conversion Factors 1 km = 0.62137 mile; 1 km = 1x103 m 7 days = 1 week Step 4: Equation meters/day x (km/m) x (mi/km) x (weeks/day) Step 5: Solve 2,100 m/day x (1 km/1x103m) x (0.62137 mi/km) x (7 days/week) 9.134 miles/week =24 mi/week (2 sig fig) Planck’s constant is 6.626x10-34 J·s; Express this constant in btu·hour Step 1: Starting Units J·s Step 2: Ending Units btu · hour Step 3: Conversion Units 1 btu = 1055.06 J 60 s = 1 min; 60 min = 1 hour Step 4: Equation J·s x (btu/J) x (s/min) x (min/hour) Step 5: Solve 6.626x10-34 J·s x (1 btu/1055.06J) x (60 s/1 min) x (60 min/1 hr) = 2.260876x10-33 btu·hour =2.261x10-33 btu·hour (4 sig fig)