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Complex Unit Conversion
Many magnitudes have a relation with more than one unit. For example, speed is measured in meters per second (m/s), density in grams per cubic centimeter (g/cm³), etc. To convert these complex units, we need to use a conversion factors for each individual unit.
Understanding Compound Units
When working with compound units like speed, remember that 100 km/h means:
\[100 \frac{ \text{km}}{ \text{h}} = \frac{100 \text{ km}}{1 \text{ h}}\]
Important: When converting time units in the denominator, the time unit must be placed on top in the conversion factor. For example, to convert hours to seconds: use 3600 s / 1 h, not 1 h / 3600 s. This ensures the unwanted unit cancels out properly.
Example of unit changes
| Desired Conversion | Equivalence Relationship | Formula and Calculation |
|---|---|---|
| 90 km/h to m/s | 1 km = 1000 m, 1 h = 3600 s |
\[90 \frac{ \text{km}}{ \text{h}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 25 \text{ m/s}\]
| | 3.5 g/cm³ to kg/m³ | 1 g = 0.001 kg, 1 cm³ = 10⁻⁶ m³ |
\[3.5 \frac{ \text{g}}{ \text{cm}^3} \times \frac{0.001 \text{ kg}}{1 \text{ g}} \times \frac{1 \text{ cm}^3}{10^{-6} \text{ m}^3} = 3500 \text{ kg/m}^3\]
| | 25 L/min to mL/s | 1 L = 1000 mL, 1 min = 60 s |
\[25 \frac{ \text{L}}{ \text{min}} \times \frac{1000 \text{ mL}}{1 \text{ L}} \times \frac{1 \text{ min}}{60 \text{ s}} = 417 \text{ mL/s}\]
| | 5 m/s² to km/h² | 1 m = 0.001 km, 1 s = 1/3600 h |
\[5 \frac{ \text{m}}{ \text{s}^2} \times \frac{0.001 \text{ km}}{1 \text{ m}} \times \frac{(3600 \text{ s})^2}{1 \text{ h}^2} = 64800 \text{ km/h}^2\]
| | 200 mg/L to g/m³ | 1 mg = 0.001 g, 1 L = 0.001 m³ |
\[200 \frac{ \text{mg}}{ \text{L}} \times \frac{0.001 \text{ g}}{1 \text{ mg}} \times \frac{1 \text{ L}}{0.001 \text{ m}^3} = 0.2 \text{ g/m}^3\]
| | 1.5 atm to Pa | 1 atm = 101325 Pa |
\[1.5 \text{ atm} \times \frac{101325 \text{ Pa}}{1 \text{ atm}} = 151988 \text{ Pa}\]
| | 120 J/min to W | 1 W = 1 J/s, 1 min = 60 s |
\[120 \frac{ \text{J}}{ \text{min}} \times \frac{1 \text{ min}}{60 \text{ s}} = 2 \text{ J/s} = 2 \text{ W}\]
| | 60 mph to km/h | 1 mile = 1.609 km |
\[60 \frac{ \text{mile}}{ \text{h}} \times \frac{1.609 \text{ km}}{1 \text{ mile}} = 96.5 \text{ km/h}\]
| | 15 N/cm² to Pa | 1 N/cm² = 10⁴ Pa |
\[15 \frac{ \text{N}}{ \text{cm}^2} \times \frac{10^4 \text{ Pa}}{1 \text{ N/cm}^2} = 1.5 \times 10^5 \text{ Pa}\]
|
Solved Examples
Example 1: Convert 0.5 km³ to cm³
Method 1: Direct conversion
1 km = 1,000 m = 100,000 cm
So 1 km³ = (100,000 cm)³ = 10¹⁵ cm³
\[0.5 \text{ km}^3 \times \frac{10^{15} \text{ cm}^3}{1 \text{ km}^3} = 5 \times 10^{14} \text{ cm}^3\]
> Method 2: Step-by-step >
\[0.5 \text{ km}^3 \times \frac{10^9 \text{ m}^3}{1 \text{ km}^3} \times \frac{10^6 \text{ cm}^3}{1 \text{ m}^3} = 0.5 \times 10^{15} \text{ cm}^3 = 5 \times 10^{14} \text{ cm}^3\]
> Result: 0.5 km³ = 5 × 10¹⁴ cm³ = 500,000,000,000,000 cm³
Example 2: A rectangular water tank is 25 m long and 12.5 m wide and 2 m high. Calculate its volume in liters and cubic feet.
Step 1: Calculate volume in m³
\[\text{Volume} = 25 \text{ m} \times 12.5 \text{ m} \times 2 \text{ m} = 625 \text{ m}^3\]
> Step 2: Convert to liters >
\[625 \text{ m}^3 \times \frac{1,000 \text{ L}}{1 \text{ m}^3} = 625,000 \text{ L}\]
> Step 3: Convert to cubic feet > 1 m³ ≈ 35.31 ft³ >
\[625 \text{ m}^3 \times \frac{35.31 \text{ ft}^3}{1 \text{ m}^3} ≈ 22,069 \text{ ft}^3\]
> Result: Tank volume = 625 m³ = 625,000 L ≈ 22,069 ft³
Example 3: A swimming pool is 15 m long and 800 cm wide and 150 cm deep. What is its volume in liters?
Method 1: Convert everything to m first
Length: 15 m
Width: 800 cm = 8 m
Depth: 150 cm = 1.5 m
\[\text{Volume} = 15 \text{ m} \times 8 \text{ m} \times 1.5 \text{ m} = 180 \text{ m}^3\]
> Method 2: Convert to liters >
\[180 \text{ m}^3 \times \frac{1,000 \text{ L}}{1 \text{ m}^3} = 180,000 \text{ L}\]
> Result: Pool volume = 180,000 L
Example 4: A cylindrical tank has a diameter of 200 cm and height 150 cm. Find its volume in liters and gallons.
Step 1: Find radius and calculate volume
Radius = 200 cm ÷ 2 = 100 cm = 1 m, Height = 150 cm = 1.5 m
\[V = \pi r^2 h = 3.14159 \times (1 \text{ m})^2 \times 1.5 \text{ m} = 4.71 \text{ m}^3\]
> Step 2: Convert to liters >
\[4.71 \text{ m}^3 \times \frac{1,000 \text{ L}}{1 \text{ m}^3} = 4,710 \text{ L}\]
> Step 3: Convert to gallons >
\[4,710 \text{ L} \times \frac{1 \text{ gal}}{3.785 \text{ L}} ≈ 1,244 \text{ gal}\]
> Result: Tank volume = 4.71 m³ = 4,710 L ≈ 1,244 gal
Example 5: A cube has sides of 20 cm. What is its volume in liters and milliliters?
Step 1: Calculate volume in cm³
\[V = (20 \text{ cm})^3 = 8,000 \text{ cm}^3\]
> Step 2: Convert to liters > Since 1 L = 1,000 cm³: >
\[8,000 \text{ cm}^3 \times \frac{1 \text{ L}}{1,000 \text{ cm}^3} = 8 \text{ L}\]
> Step 3: Convert to milliliters > Since 1 cm³ = 1 mL: >
\[8,000 \text{ cm}^3 = 8,000 \text{ mL}\]
> Result: Cube volume = 8,000 cm³ = 8 L = 8,000 mL