Problem 58 - Physics for Scientists and Engineers with Modern Physics

Question

A watch manufacturer claims that its watches gain or lose no more than 8 seconds in a year. How accurate is this watch, expressed as a percentage?

Step-by-Step Solution

Answer

The watch is accurate to approximately 0.0000254%.

Step 1: Understand the Problem

The manufacturer claims the watch gains or loses no more than 8 seconds in a year. We need to find out how this compares to the total number of seconds in a year, as a percentage.

Step 2: Calculate Seconds in a Year

First, calculate the total number of seconds in a year. There are 365 days in a year, 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute. Thus, calculate the total seconds as follows: \( 365 \times 24 \times 60 \times 60 \).

Step 3: Calculate Total Seconds

Perform the calculation: \[ 365 \times 24 = 8760 \]\[ 8760 \times 60 = 525600 \]\[ 525600 \times 60 = 31536000 \]Thus, there are 31,536,000 seconds in a year.

Step 4: Determine Deviation as a Percentage

Now, calculate the deviation of 8 seconds as a percentage of the total seconds in a year: \[ \frac{8}{31536000} \times 100 \approx 0.0000254\% \]

Step 5: Interpret the Result

The percentage deviation of 8 seconds per year is approximately 0.0000254%. This is an extremely small deviation, indicating high accuracy in the watch's timekeeping.

Key Concepts

Seconds in a Year

When considering how accurate a watch is, we start by examining the total number of seconds in a year. Since time is measured in seconds, it's important to get this number right. To calculate the seconds in a year, follow these steps:

• There are 365 days in a standard year.
• Each day has 24 hours.
• Each hour consists of 60 minutes.
• Every minute contains 60 seconds.

Multiply these numbers together to find the total seconds: \[365 \times 24 \times 60 \times 60 = 31,536,000\]This resulting product, 31,536,000 seconds, allows us to compare any time deviations — like those of a watch — effectively in terms of accuracy.

Percentage Calculation

To evaluate the precision of a watch's timekeeping, we need to express its deviation as a percentage of the total seconds in a year. Percentage calculation gives us a way to represent this small deviation and understand its relative size.Let's break down the calculation:

• Assume the watch gains or loses a maximum of 8 seconds per year.
• The total seconds in a year are 31,536,000, as calculated earlier.

We now find the percentage by taking the ratio of the deviation to the total seconds and multiplying by 100:\[\left(\frac{8}{31,536,000}\right) \times 100\]This gives us an approximate deviation of 0.0000254%. Such a tiny percentage reflects the impressive accuracy of time measurement provided by the watch.

Time Deviation

Time deviation is a measure of how much the actual timekeeping of a watch diverges from perfect time over a given period; in this case, a year. Understanding deviation is crucial for appreciating a watch's precision. For the watch in question:

• It deviates by no more than 8 seconds in one year.
• This deviation is negligible when compared to the 31,536,000 seconds in a year.

The smaller the deviation, the more accurate the timepiece. In our situation, a deviation of just 8 seconds across an entire year is remarkable. It underscores the advanced engineering and meticulous design efforts that ensure minimal time loss or gain.

Watch Precision

Watch precision refers to the ability of a watch to keep time closely aligned to a designated standard without gaining or losing time excessively. Precision is critical for those relying on accurate timekeeping, whether for personal scheduling or professional needs. A precise watch:

• Gains or loses only a minimal amount of time over long periods.
• Reflects careful manufacturing and design considerations.

In our example, the claimed precision, with potential deviation capped at 8 seconds annually, is extraordinary. It indicates that for every million seconds, the watch deviates by only a fraction, showing how trustworthy and dependable such a device can be for its users. Watch precision assures that time can be managed effectively and reliably, contributing to better time management and planning.