Problem 73 - Master Resource Book in JEE Main Physics

Question

There are atomic clocks capable of measuring time with an accuracy of 1 part in \(10^{11} .\) If two such clocks are operated with precision, then after running for \(5000 \mathrm{yr}\), these will record (a) a difference of nearly \(2 \mathrm{~s}\) (b) a difference of 1 day (c) a difference of \(10^{11} \mathrm{~s}\) (d) a difference of \(1 \mathrm{yr}\)

Step-by-Step Solution

Answer

(a) a difference of nearly 2 seconds

Step 1: Understand the Time Measurement Accuracy

The atomic clocks have an accuracy of 1 part in \(10^{11}\). This means that over a large time period, the error in measurement will only be \(1\) part out of \(10^{11}\) parts.

Step 2: Calculate Total Seconds in 5000 Years

First, convert 5000 years into seconds. Each year has about \(365.25\) days (considering leap years), each day has \(24\) hours, each hour has \(3600\) seconds. So:\[5000 \times 365.25 \times 24 \times 3600 = 1.57788 \times 10^{11} \text{ seconds}\]

Step 3: Determine Measurement Error Over 5000 Years

Given the clock's precision of 1 part in \(10^{11}\), the error after \(1.57788 \times 10^{11}\) seconds is:\[\frac{1.57788 \times 10^{11}}{10^{11}} = 1.57788 \text{ seconds}\]

Step 4: Conclusion Based on Calculations

After calculating, the error or difference in time measured by the atomic clocks over \(5000\) years will be approximately \(1.57788\) seconds. So the correct choice is that the difference is nearly \(2\) seconds.

Key Concepts

Time Measurement Accuracy

Time measurement accuracy is an essential aspect of any clock, especially for sophisticated devices like atomic clocks. These clocks operate with an incredible degree of precision. Time measurement accuracy, in this context, is expressed as 1 part in \(10^{11}\). This means for every \(10^{11}\) parts of time measured, only one part might be slightly off.

Such precision ensures that these clocks remain reliable over long periods, only accumulating minor errors.

• This level of accuracy allows us to effectively gauge long intervals of time, making atomic clocks crucial in technological applications that rely on perfect timing.
• Atomic clocks are useful for global positioning systems (GPS), telecommunications, and scientific research.

Atomic clocks' sophisticated functional mechanisms provide benchmarks for subsequent developments in timekeeping technologies.

Measurement Error Calculation

Calculating measurement error is critical to understanding how precise these timekeeping devices are over lengthy periods. For atomic clocks with an accuracy of 1 part in \(10^{11}\), it is significant to determine how much error builds over time.

In a scenario where such a clock operates for \(5000\) years, let's break it down:

1. **Determine Total Time in Seconds**: Convert \(5000\) years into seconds: - There are approximately \(365.25\) days per year (accounting for leap years). - Each day has \(24\) hours, and each hour has \(3600\) seconds. - Therefore, the total time in seconds is: \[5000 \times 365.25 \times 24 \times 3600 = 1.57788 \times 10^{11} \text{ seconds}\] 2. **Calculate Error Over Time**: Given the accuracy of 1 part in \(10^{11}\), calculate potential error: - The potential error is: \[\frac{1.57788 \times 10^{11}}{10^{11}} = 1.57788 \text{ seconds}\]The computed error indicates that even over \(5000\) years, these clocks remain staggeringly precise, with an expected error of only around \(1.57788\) seconds.

Atomic Clock Precision

Atomic clock precision sets the standard for modern timekeeping. These clocks employ the oscillations of atoms, such as cesium or rubidium, to generate a consistent time interval.

The advantageous precision arises because atomic frequencies are far less affected by environmental changes than mechanical or quartz clocks. Each tick is generated by the oscillation of atoms at stable frequencies, maintaining a reliable measure of time.

• Atomic clocks remain the backbone of international timekeeping standards and scientific explorations.
• Their precision is why they serve vital roles in fields like astronomy and GPS technology.

Atomic clocks' development marked a significant leap in enabling accurate synchronization across diverse systems, ensuring reliable global communication and navigation.