Problem 24 - Physics for Scientists and Engineers with Modern Physics

Question

(1) Estimate the order of magnitude (power of ten) of: \((a) 2800\) , (b) \(86.30 \times 10^{2},(c) 0.0076,\) and \((d) 15.0 \times 10^{8} .\)

Step-by-Step Solution

Answer

(a) 10^3, (b) 10^3, (c) 10^{-3}, (d) 10^9.

Step 1: Understanding Order of Magnitude

The order of magnitude of a number is essentially the power of 10 when the number is expressed in scientific notation. For example, for a number \( \times 10^n \), the order of magnitude is \( n \).

Step 2: Estimating the Order of Magnitude for 2800

Convert 2800 to scientific notation as \( 2.8 \times 10^3 \). The order of magnitude is \(10^3\).

Step 3: Estimating the Order of Magnitude for 86.30 x 10²

Convert \(86.30 \times 10^2\) to scientific notation as \(8.63 \times 10^3 \). The order of magnitude is \(10^3\).

Step 4: Estimating the Order of Magnitude for 0.0076

Convert 0.0076 to scientific notation as \(7.6 \times 10^{-3} \). The order of magnitude is \(10^{-3}\).

Step 5: Estimating the Order of Magnitude for 15.0 x 10⁸

Convert \(15.0 \times 10^8\) to scientific notation as \(1.5 \times 10^9 \). The order of magnitude is \(10^9\).

Key Concepts

Scientific Notation

Scientific notation is a method used to express large or small numbers in a more manageable form. In this system, numbers are represented as a product of a number between 1 and 10 and a power of ten. This makes it easier to understand and compare numbers, especially when dealing with very large or very small quantities.

For example, the number 2800 can be written as \( 2.8 \times 10^3 \). This is a compact way of writing the number without all the zeros you would normally need.

Similarly, a smaller number like 0.0076 can be expressed as \( 7.6 \times 10^{-3} \), removing the leading zeros.

This notation is especially beneficial in scientific fields where such numbers occur frequently.

• It simplifies calculations involving very large or small numbers.
• Facilitates easy comparison of different scales of numbers.
• Makes the order of magnitude (or scale) of the number clear at a glance.

Power of Ten

The power of ten in scientific notation helps us understand the scale or size of the number. The power indicates how many times we need to multiply 10 to reach the number.

When you see a number like \( 10^3 \), it means 10 is multiplied by itself 3 times, which equals 1000. A negative power such as \( 10^{-3} \) means you are dividing by 10 three times, resulting in 0.001.

This makes it easy to work with numbers without counting loads of zeros, which is particularly useful in scientific calculations.

• The higher the power, the larger the number.
• A negative power represents a small fractional number.
• This method provides quick insights into the size and scale of a number.

Estimation

Estimation is a crucial skill, especially when precise calculations are unnecessary or impractical. It's about finding an approximate value that is close enough to the real number for the context of a problem.

Estimating using the order of magnitude involves rounding a number to the nearest power of ten.

• For 2800, which is closer to \( 10^3 \), we convert it to \( 2.8 \times 10^3 \), simplifying to \( 10^3 \).
• Similarly, 86.30 combined with \( 10^2 \) becomes \( 8.63 \times 10^3 \), rounding internally to emphasize the scale as \( 10^3 \).
• A tiny number like 0.0076 translates to \( 7.6 \times 10^{-3} \), making its scale clear with \( 10^{-3} \).
• And larger numbers like \( 15.0 \times 10^8 \) simplify to \( 1.5 \times 10^9 \) reflecting \( 10^9 \).

This process helps quickly assess if more complex calculations are necessary and is a fundamental tool in both academic exercises and real-world applications.