Problem 10 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

(a) 88 ft/s, (b) 9.754 m/s², (c) 1000 kg/m³.

Step 1: Convert MPH to FPS

To convert the speed from miles per hour (mph) to feet per second (fps), you need to use the conversion factors: 1 mile = 5280 feet and 1 hour = 3600 seconds. Start with 60 mph and convert to fps by using the formula: \[ 60 \text{ mph} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = 88 \text{ ft/s}. \] Therefore, 60 mph is equal to 88 ft/s.

Step 2: Convert Acceleration from ft/s² to m/s²

To convert 32 ft/s² to m/s², use the conversion factor: 1 ft = 30.48 cm, and then convert centimeters to meters (1 m = 100 cm). The conversion is done as follows: \[ 32 \text{ ft/s}^2 \times \frac{30.48 \text{ cm}}{1 \text{ ft}} \times \frac{1 \text{ m}}{100 \text{ cm}} = 9.754 \text{ m/s}^2. \] Thus, the acceleration is approximately 9.754 m/s².

Step 3: Convert Density from g/cm³ to kg/m³

To convert density from g/cm³ to kg/m³, use the conversion factors: 1 g = 0.001 kg and 1 cm³ = 1e-6 m³. Therefore, the conversion is: \[ 1.0 \text{ g/cm}^3 \times \frac{1 \text{ kg}}{1000 \text{ g}} \times \frac{1,000,000 \text{ cm}^3}{1 \text{ m}^3} = 1000 \text{ kg/m}^3. \] Hence, the density of water is 1000 kg/m³.

Key Concepts

Velocity Conversion

Converting velocity from one unit to another is a common task in physics. Let’s explore how to convert from miles per hour (mph) to feet per second (fps), a useful skill for many science classes. When you start with a velocity of 60 mph and need to convert to fps, there are two key conversion factors to remember:

• 1 mile = 5280 feet
• 1 hour = 3600 seconds

First, multiply 60 by 5280 to convert miles to feet. This gives us the number of feet traveled in one hour. Next, divide by 3600 to convert hours to seconds. Therefore, the conversion can be expressed as:\[60 \text{ mph} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ s}} = 88 \text{ ft/s}\]This formula shows that 60 mph is equal to 88 feet per second. Understanding this conversion is important for situations where speed measurements need precision on a smaller scale than miles.

Acceleration Conversion

Acceleration conversion is another essential concept, especially when dealing with different measurement systems. For example, let's change the unit from feet per second squared (ft/s²) to the standard SI units of meters per second squared (m/s²). Acceleration of a freely falling object is commonly given as 32 ft/s².To convert this, you’ll use the following relationships:

• 1 foot = 30.48 centimeters
• 1 meter = 100 centimeters

So, you start by converting feet to centimeters and then centimeters to meters. The conversion is as follows:\[32 \text{ ft/s}^2 \times \frac{30.48 \text{ cm}}{1 \text{ ft}} \times \frac{1 \text{ m}}{100 \text{ cm}} = 9.754 \text{ m/s}^2\]This means that the acceleration is approximately 9.754 m/s². Converting between these units helps communicate scientific concepts across global standards.

Density Conversion

Converting density units enables better comprehension between different scientific studies. Let’s see how to turn a density value from grams per cubic centimeter (g/cm³) to kilograms per cubic meter (kg/m³).For water, the density is known as 1.0 g/cm³. According to the unit conversion rates:

• 1 gram = 0.001 kilograms
• 1 cubic centimeter = 1e-6 cubic meters

Apply these conversions:\[1.0 \text{ g/cm}^3 \times \frac{1 \text{ kg}}{1000 \text{ g}} \times \frac{1,000,000 \text{ cm}^3}{1 \text{ m}^3} = 1000 \text{ kg/m}^3\]Thus, the density of water calculates to 1000 kg/m³. This conversion makes it easier to compare substances when using the metric system, which is the most widely used system of measurement in scientific research. Understanding these conversions help in applying density values correctly in volumetric calculations.