Problem 10 - University Physics with Modern Physics

Question

The following conversions occur frequently in physics and are very useful. (a) Use 1 mi \(=5280 \mathrm{ft}\) and \(1 \mathrm{h}=3600 \mathrm{s}\) to convert 60 \(\mathrm{mph}\) to units of \(\mathrm{ff} / \mathrm{s} .(\mathrm{b})\) The acceleration of a freely falling object is 32 \(\mathrm{ff} / \mathrm{s}^{2} .\) Use \(1 \mathrm{ft}=30.48 \mathrm{cm}\) to express this acceleration in units of \(\mathrm{m} / \mathrm{s}^{2} .\) (c) The density of water is 1.0 \(\mathrm{g} / \mathrm{cm}^{3} .\) Convert this density to units of \(\mathrm{kg} / \mathrm{m}^{3} .\)

Step-by-Step Solution

Answer

(a) 60 mph = 88 ft/s; (b) 32 ft/s² = 9.75 m/s²; (c) 1.0 g/cm³ = 1000 kg/m³.

Step 1: Convert miles per hour to feet per second

We need to convert 60 miles per hour (mph) to feet per second (ft/s). Start by breaking down the conversions needed:\[1 \text{ mile} = 5280 \text{ ft} \]\[1 \text{ hour} = 3600 \text{ seconds} \]Thus, to convert 60 mph to ft/s:\[60 \text{ mph} = 60 \times \frac{5280 \text{ ft}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ s}} \]Simplifying, we calculate:\[60 \times \frac{5280}{3600} = \frac{60 \times 5280}{3600} \approx 88 \text{ ft/s} \]

Step 2: Convert acceleration to meters per second squared

The acceleration of a freely falling object is given as 32 ft/s². To convert this to meters per second squared (m/s²), we use:\[1 \text{ ft} = 30.48 \text{ cm} = 0.3048 \text{ m} \]Therefore, the acceleration in m/s² is:\[32 \text{ ft/s}^2 = 32 \times 0.3048 \text{ m/s}^2 \approx 9.75 \text{ m/s}^2 \]

Step 3: Convert density to kilograms per cubic meter

The density of water is given as 1.0 g/cm³. We need to convert it to kg/m³. Knowing:\[1 \text{ g/cm}^3 = 1000 \text{ kg/m}^3 \]Thus, simply convert the units:\[1.0 \text{ g/cm}^3 = 1.0 \times 1000 \text{ kg/m}^3 = 1000 \text{ kg/m}^3 \]

Key Concepts

Conversion of Speed Units

Understanding how to convert speed units is essential as different regions of the world might use different measurement systems. To convert speed from miles per hour (mph) to feet per second (ft/s), you need to know two key facts about the units:

• 1 mile is equal to 5280 feet.
• 1 hour contains 3600 seconds.

Let's take an example: converting 60 mph to ft/s. The equation becomes: \[60 ext{ mph} = 60 \times \frac{5280 \text{ ft}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ s}}\]By simplifying the multiplication, you will yield: \[60 \times \frac{5280}{3600} = 88 \text{ ft/s}\]Thus, when you drive at 60 mph, you can think of it as moving 88 feet every second.
Converting speed helps you understand how fast objects are moving in different measurement systems, which is valuable in fields like physics and engineering.

Conversion of Acceleration Units

Acceleration tells us how quickly an object's speed changes. It can be helpful to convert these numbers into different units based on the conditions or systems you’re working with. In physics, converting acceleration from feet per second squared (ft/s²) to meters per second squared (m/s²) is a common task. Remember:

• 1 foot equals 30.48 centimeters, which is 0.3048 meters.
• Acceleration given: 32 ft/s².

To convert, use the formula:\[32 \text{ ft/s}^2 = 32 \times 0.3048 \text{ m/s}^2 \]Calculate to get:\[ 32 \times 0.3048 \approx 9.75 \text{ m/s}^2 \]This conversion is particularly useful in scientific calculations and engineering as it aligns with the metric system, widely used in technical and educational fields around the globe.

Conversion of Density Units

Density is a measure of how much mass is contained in a given volume. In laboratories or industry settings, converting the density of substances into different units is crucial for accurate measurements and comparisons. Density conversions often involve changing from grams per cubic centimeter (g/cm³) to kilograms per cubic meter (kg/m³), as the latter is more commonly used in scientific contexts.For instance, the density of water is 1.0 g/cm³. To convert it using the key conversion factor:

• 1 g/cm³ = 1000 kg/m³.

Simply apply the calculation:\[1.0 \text{ g/cm}^3 = 1.0 \times 1000 \text{ kg/m}^3 = 1000 \text{ kg/m}^3\]This tells us that the density of water is 1000 kg/m³, making it easy to compare with other materials and essential for computations in fluid dynamics or chemistry.