Question

A convenient substitution for the number of seconds in a year is \(\pi\) times \(10^{7}\). To within what percentage error is this correct?

Short answer

The approximate substitution is off by about 0.38% from the accurate value of the seconds in a year.

Step-by-step solution

Determine the accurate number of seconds in a year

We can calculate the accurate number of seconds in a year by multiplying the number of seconds in a minute (60) by the number of minutes in an hour (60), the number of hours in a day (24), and the number of days in a year (365). This amounts to \(60 * 60 * 24 * 365 = 31536000\) seconds.

Analyze the given substitution value

The given substitution value for the number of seconds in a year is \(\pi * 10^{7}\). So the estimated number of seconds in a year is approximately \(\pi * 10^{7} = 31415926.5\) seconds.

Find the difference between the accurate and estimated numbers

The difference between the accurate number of seconds in a year and the estimated substitution value is \(31536000 - 31415926.5 = 120073.5\) seconds.

Convert the difference into a percentage

Percentage error = (difference between accurate and estimated values / accurate value) * 100. Thus, by substituting the values: Percentage error = \((120073.5 / 31536000) * 100 = 0.38%\) approximately.

Interpret the result

This means the substitution approximation is off by around 0.38% from the accurate value of seconds in a year.

Key concepts

Time Measurement

Time measurement is the process of quantifying the passage of time. It is an integral aspect of everyday life, scientific studies, and industrial applications. The standard unit of time in the International System of Units (SI) is the second. One year, for example, is traditionally determined by the Earth's revolutions around the Sun and contains approximately 365 days. By breaking down this period into smaller units, like days into hours, hours into minutes, and minutes into seconds, we obtain an approximate value of seconds in a year: 60 seconds per minute, 60 minutes per hour, 24 hours per day, and 365 days a year lead to a calculation of 31,536,000 seconds in a standard year.

Despite the simplicity of this computation, it's important to note that a precise measurement of time can be complex due to factors such as leap years, leap seconds, and Earth's variable orbital speed. Therefore, while we can perform these calculations using average values for practical purposes, exact timekeeping systems, like atomic clocks, are necessary for high precision requirements in fields such as navigation, astronomy, and communication.

Scientific Approximation

Scientific approximation is a mathematical process used to simplify complex numbers, making them easier to work with while maintaining a level of accuracy that is acceptable for a given purpose. In the context of our exercise, the number of seconds in a year has been approximated using the number \(\pi \times 10^{7}\), which is a memorable and conveniently rounded-off number. Such approximations are common in science and engineering to facilitate quick calculations.

It's important for students to understand that all scientific approximations come with a degree of uncertainty, referred to as an error. This is calculated as the percentage error, which quantifies how close an approximation is to the true or accepted value. In practical scenarios, the choice of approximation and the level of acceptable error depends on the requirements of the task at hand. The chosen approximation of \(\pi \times 10^{7}\) seconds for one year provides a balance between ease of use and level of accuracy, which in most cases is sufficient for rough calculations or as a mnemonic aid.

Dimensional Analysis

Dimensional analysis is a tool that helps in verifying the physical consistency of equations, converting units, and solving problems. When applied to the exercise, we can see how the conversion factors (60 seconds in a minute, 60 minutes in an hour, etc.) are used to systematically transition from days to seconds. This method ensures that the final answer is in the desired units of seconds.

One of the crucial aspects of dimensional analysis is the concept of unity bracket which converts one unit to another by multiplying with a factor that equals one unit of the desired measure. For example, \(60 \text{ minutes/hour}\times 60 \text{ seconds/minute}\) uses the unity bracket to convert hours to seconds. Dimensional analysis not only helps students to convert units but also to comprehend the relationship between different physical quantities, thus bolstering their understanding of fundamental concepts in physics and engineering.