Problem 11 - Physics for Scientists and Engineers with Modern Physics

Question

$$ \begin{array}{l}{\text { (1) Write the following as full (decimal) numbers with stan- }} \\ {\text { dard units: }(a) 286.6 \mathrm{mm},(b) 85 \mu \mathrm{V},(c) 760 \mathrm{mg},(d) 60.0 \mathrm{ps}} \\ {(e) 22.5 \mathrm{fm},(f) 2.50 \text { gigavolts. }}\end{array} $$

Step-by-Step Solution

Answer

(a) 0.2866 m, (b) 0.000085 V, (c) 0.76 g, (d) 0.00000000006 s, (e) 0.0000000000000225 m, (f) 2,500,000,000 V.

Step 1: Identify the Metric Prefixes

Each measurement unit has a prefix that represents a certain factor of 10. Recognize the following prefixes:- 'mm' stands for millimeters, which is \(10^{-3}\) meters.- 'μV' stands for microvolts, which is \(10^{-6}\) volts.- 'mg' stands for milligrams, which is \(10^{-3}\) grams.- 'ps' stands for picoseconds, which is \(10^{-12}\) seconds.- 'fm' stands for femtometers, which is \(10^{-15}\) meters.- 'giga' in gigavolts represents \(10^9\) volts.

Step 2: Convert and Write in Standard Units (a-b)

Convert each measurement to decimal form in standard base units:- (a) \(286.6 \, \text{mm}\) becomes \(286.6 \times 10^{-3} \, \text{m} = 0.2866 \, \text{m}\)- (b) \(85 \, \mu\text{V}\) becomes \(85 \times 10^{-6} \, \text{V} = 0.000085 \, \text{V}\)

Step 3: Convert and Write in Standard Units (c-d)

Continue converting the next set of measurements:- (c) \(760 \, \text{mg}\) becomes \(760 \times 10^{-3} \, \text{g} = 0.76 \, \text{g}\)- (d) \(60.0 \, \text{ps}\) becomes \(60.0 \times 10^{-12} \, \text{s} = 0.00000000006 \, \text{s}\)

Step 4: Convert and Write in Standard Units (e-f)

Convert the remaining measurements:- (e) \(22.5 \, \text{fm}\) becomes \(22.5 \times 10^{-15} \, \text{m} = 0.0000000000000225 \, \text{m}\)- (f) \(2.50 \, \text{gigavolts}\) becomes \(2.50 \times 10^9 \, \text{V} = 2,500,000,000 \, \text{V}\)

Key Concepts

Standard Units

Standard units play a crucial role in ensuring consistency and clarity in measurements worldwide. These units, also known as the International System of Units (SI), provide a clear and standardized way to express physical quantities such as length, mass, time, and temperature.

Some common standard units include:

• Meter (m) for length
• Second (s) for time
• Kilogram (kg) for mass
• Litre (L) for volume
• Kelvin (K) for temperature

Using standard units helps in operations like addition and multiplication, as the units are consistent and universally recognized. For example, when you convert 286.6 millimeters to meters, you are aligning it with the global standard, making it easier to use in scientific calculations.

Decimal Conversion

Decimal conversion is the process of transforming a value with a metric prefix into its full decimal form. Metric prefixes denote powers of ten, making it easy to express both large and small quantities concisely. When converting, it's essential to replace the metric prefix with the correct power of ten.

Metric prefixes include:

• Milli (m) = \(10^{-3}\)
• Micro (μ) = \(10^{-6}\)
• Giga (G) = \(10^{9}\)
• Pico (p) = \(10^{-12}\)
• Femto (f) = \(10^{-15}\)

For example, converting 85 microvolts (μV) to volts involves multiplying 85 by \(10^{-6}\), which equates to 0.000085 volts. Such conversions ensure clarity in measurements without ambiguity.

Measurement Units

Measurement units are tools that provide a way to quantify physical properties, facilitating scientific and everyday calculations. Each measurement unit corresponds to a particular kind of physical quantity. By using precise measurement units alongside metric prefixes, we can measure, record, and communicate values accurately.

Examples of measurements include:

• Length measured in meters (m)
• Voltage measured in volts (V)
• Mass measured in grams (g)

For instance, converting 2.50 gigavolts to volts reflects the measurement unit (volt) and applies the giga prefix correctly, resulting in a value of 2,500,000,000 volts. Understanding and accurately using measurement units and metric prefixes are foundational skills in science, enhancing both precision and comprehension.