Problem 1 - Physics for Scientists and Engineers with Modern Physics

Question

(1) The age of the universe is thought to be about 14 billion years. Assuming two significant figures, write this in powers of ten in \((a)\) years, \((b)\) seconds.

Step-by-Step Solution

Answer

(a) The age is \(1.4 \times 10^{10}\) years. (b) It is \(4.4 \times 10^{17}\) seconds.

Step 1: Understanding Significant Figures

We are provided the age of the universe as 14 billion years, where the number 14 has two significant figures. Our task is to express this number using powers of ten.

Step 2: Convert to Powers of Ten for Years

First, express 14 billion in standard numerical form: 14 billion = 14,000,000,000. Now convert it to a number with powers of ten. Since the number 14,000,000,000 can be written as \(1.4 \times 10^{10}\), this is the representation with two significant figures.

Step 3: Convert Years to Seconds

To convert years into seconds, we use the fact that there are approximately 31,536,000 seconds in a year. Therefore, we multiply the age of the universe in years by this number of seconds per year.

Step 4: Calculate Seconds in Powers of Ten

Multiply \(1.4 \times 10^{10}\) years by \(31,536,000\) seconds/year. First express \(31,536,000\) as powers of ten, which is approximately \(3.1536 \times 10^{7}\). Thus, \(1.4 \times 10^{10} \times 3.1536 \times 10^{7} = 4.41 \times 10^{17}\). Round to two significant figures to keep consistent with the original data, so we get \(4.4 \times 10^{17}\) seconds.

Key Concepts

Powers of Ten

The concept of "Powers of Ten" is fundamental in mathematics, especially in representing very large or very small numbers in a compact form. Here's how it works:

• The power of ten is a way to express numbers that are either much larger or much smaller than usual for ease of reading and calculation.
• When you see an expression like \(10^n\), it means that ten is multiplied by itself \(n\) times.
• For example, \(10^3 = 10 \times 10 \times 10 = 1000\) and \(10^{-3} = \frac{1}{1000} = 0.001\).

To express 14 billion using powers of ten, you write it as \(1.4 \times 10^{10}\). This is because the number 1.4 is adjusted by moving the decimal point 10 places to the right to become 14,000,000,000. This technique is especially useful for large figures such as the age of the universe, which is approximately 14 billion years, or \(1.4 \times 10^{10}\) in powers of ten.

Conversion of Units

Converting units is a method used to change measurements from one unit to another while maintaining the same quantity. This process is essential in scientific calculations where consistency in units is required. Let’s look at how this applies:

• When we know the conversion factor, we multiply to switch from one unit to another. For instance, there are approximately 31,536,000 seconds in one year.
• To convert the age of the universe from years to seconds, multiply the number of years by this conversion factor.

For the universe's age, calculated as \(1.4 \times 10^{10}\) years, we convert to seconds by multiplying:\[1.4 \times 10^{10} \text{ years} \times 31,536,000 \text{ seconds/year} = 4.41 \times 10^{17}\text{ seconds} \] This result allows us to express time elapsed in a more precise and globally understood unit: seconds.

Scientific Notation

Scientific notation is a method used in mathematics to write numbers that are too large or too small to conveniently write in decimal form. It allows easy comparison and computation, enhancing clarity in scientific writing. Let’s delve into this:

• In scientific notation, numbers are typically written in the form \(a \times 10^n\), where \(1 \leq a < 10\) and \(n\) is an integer.
• This format is handy for maintaining significant figures, as it naturally highlights why \(a\) contains the significant figures of the measurement.

Take the example of expressing 14 billion. In scientific notation, it becomes \(1.4 \times 10^{10}\). Similarly, when the universe's age is converted from years to seconds, it is \(4.4 \times 10^{17}\) seconds after rounding to two significant figures. Scientific notation helps simplify the handling of these large figures, keeping calculations manageable and interpretations clear.