Problem 41 - Physics for Scientists and Engineers with Modern Physics

Question

(a) How many scconds are there in 1.00 year? (b) How many nanoseconds are there in 1.00 year? (c) How many years are there in 1.00 second?

Step-by-Step Solution

Answer

(a) 31,536,000 seconds, (b) 3.1536 x 10^16 nanoseconds, (c) 3.1688 x 10^-8 years.

Step 1: Convert Years to Days

Start by recognizing that a year is typically considered as 365 days. This forms the base of our calculation: \[1 ext{ year} = 365 ext{ days}\]

Step 2: Convert Days to Hours

Knowing there are 24 hours in a day, convert the days into hours: \[365 ext{ days} imes 24 ext{ hours per day} = 8760 ext{ hours}\]

Step 3: Convert Hours to Minutes

Convert the number of hours into minutes since 1 hour contains 60 minutes:\[8760 ext{ hours} imes 60 ext{ minutes per hour} = 525600 ext{ minutes}\]

Step 4: Convert Minutes to Seconds

Now convert the total minutes into seconds, knowing there are 60 seconds in a minute:\[525600 ext{ minutes} imes 60 ext{ seconds per minute} = 31536000 ext{ seconds}\]

Step 5: Answer for Part (a)

Thus, there are 31,536,000 seconds in 1.00 year.

Step 6: Convert Seconds to Nanoseconds

To find the number of nanoseconds, multiply the number of seconds by the number of nanoseconds in a second (1 second = \(10^9\) nanoseconds):\[31536000 ext{ seconds} imes 10^9 ext{ nanoseconds per second} = 3.1536 imes 10^{16} ext{ nanoseconds}\]

Step 7: Answer for Part (b)

Thus, there are \(3.1536 \times 10^{16}\) nanoseconds in 1.00 year.

Step 8: Convert Seconds to Years

To find out how many years are there in one second, take the reciprocal of seconds per year:\[1 ext{ second} = \frac{1}{31536000} ext{ years}\]

Step 9: Simplify for Part (c)

This gives us the tiny fraction of a year that one second represents:\[1 ext{ second} = 3.1688 imes 10^{-8} ext{ years}\]

Step 10: Answer for Part (c)

Hence, in 1.00 second, there are approximately \(3.1688 \times 10^{-8}\) years.

Key Concepts

Time Conversion

Time conversion is a fundamental aspect of dealing with various units of time, such as seconds, minutes, hours, and years. Understanding how these units relate to each other is crucial in science and daily life. Here's a simple breakdown of basic time conversions:

• 60 seconds make up 1 minute.
• 60 minutes make up 1 hour.
• 24 hours make up 1 day.
• Typically, 365 days make up 1 year. This number can be 366 in leap years, but for general purposes, we use 365 days.

Breaking down complex periods into these simple units lets us convert between different units seamlessly. For example, converting years to seconds involves sequentially converting years to days, days to hours, hours to minutes, and finally, minutes to seconds. Each step keeps the calculations organized and simple.

Seconds to Nanoseconds

The conversion from seconds to nanoseconds involves multiplying by a large number since nanoseconds are much smaller units of time. One second is equivalent to one billion nanoseconds.To make this conversion:

• Recognize the equivalence: 1 second = \(10^9\) nanoseconds.
• If you have multiple seconds, multiply each second by \(10^9\).
• This calculation involves understanding scientific notation, which is a method of expressing large numbers succinctly.

Using scientific notation simplifies dealing with such large conversions, allowing for much more manageable calculations and reducing errors that may arise from handling so many zeros.

Seconds to Years

Converting seconds to years requires recognizing how a single second fits into a full year. This conversion is essentially the inverse of converting years to seconds.Here's a concise approach:

• Calculate the number of seconds in a year first. We've established this as 31,536,000 seconds in a typical year.
• Understand that to convert from seconds to years, you calculate the reciprocal: \(1 \text{ second} = \frac{1}{31,536,000} \text{ years}\).
• This gives the result in a very small decimal or in scientific notation: approximately \(3.1688 \times 10^{-8}\) years.

This conversion tells us how infinitesimally small a single second is compared to a full year. It's an important operation in fields like astronomy and physics, where precise time measurement is crucial.

Dimensional Analysis

Dimensional analysis is a critical technique used in solving problems involving unit conversions. It helps ensure calculations are consistent and correct by keeping track of the units through multiplication or division. When performing a conversion:

• Write down the conversion factors that will multiply your original quantity. These factors are equivalences like 1 inch = 2.54 centimeters, or in this context, 1 year = 31,536,000 seconds.
• Use these conversion factors in a way that allows the unwanted units to cancel out, leaving the desired units.
• This technique is excellent for preventing and catching errors because each factor must cancel correctly.

Dimensional analysis helps in systematically approaching conversion problems, making it easier to understand complex relationships between different measurement systems.