Q.Which of the following time measuring devices is most precise?

Precision depends on an instrument's least count (smallest interval it can detect):

• Wall clock: $\approx 1\ \text{s}$
• Stopwatch: $\approx 0.1$–$0.01\ \text{s}$
• Digital watch: $\approx 0.01\ \text{s}$ …

Comparing the four devices by least count

1. Wall clock — typically has only a second hand; least count $\approx 1\ \text{s}$. Cannot resolve fractions of a second.
2. Stop watch — mechanical or digital, least count typically $0.1\ \text{s}$ or $0.01\ \text{s}$.
3. Digital watch — usually displays to $0.01\ \text{s}$, better than a mechanical stopwatch but still far from atomic precision.
4. Atomic clock — uses the fixed frequency of a specific atomic transition (e.g. the cesium-133 hyperfine transition, which defines the SI second) as its "tick." This makes it accurate to about $1$ part in $10^{13}$ — an uncertainty of roughly $10^{-13}\ \text{s}$, meaning it would gain or lose only a few microseconds over an entire year. …

Method: Least Count Comparison

Method name: Least Count (or Resolution) Comparison

Concept: The precision of a measuring device is determined by its least count — the smallest change in the measured quantity that the device can detect. A smaller least count means higher precision.

Steps to solve:

1. Identify the least count of each device

   • Wall clock: typically measures up to 1 second (least count = 1 s)
   • Stopwatch: usually measures up to 0.1 s or 0.01 s (least count = 0.1 s or 0.01 s)
   • Digital watch: commonly shows up to 0.01 s (least count = 0.01 s)
   • Atomic clock: accurate to about 1 part in $10^{13}$ — an uncertainty of order $10^{-13}$ s (from the fixed cesium-133 hyperfine transition frequency that defines the SI second)

2. Compare the least counts

   Smaller least count → more precise device.

3. Rank them in increasing order of least count

   • Atomic clock: $\sim 10^{-13}$ s (smallest) …

Here are the common mistakes students make on this question, along with how to avoid each one.

Mistake 1: Confusing "Precision" with "Accuracy"

• What students do: They think a wall clock is "precise" because it shows the correct time every day, or they pick a stopwatch because it "seems scientific."
• Why it's wrong: Precision is about the smallest change a device can detect (least count), not whether it tells the correct time. A wall clock has a least count of 1 second (or more), while an atomic clock is accurate to about 1 part in $10^{13}$ — an uncertainty of order $10^{-13}$ seconds or smaller.
• How to avoid: Always define least count first. The most precise device is the one with the smallest least count.

Key rule: Precision = Least Count. Smaller least count = more precise.

Mistake 2: Ignoring the "Least Count" of a Digital Watch

• What students do: They pick a digital watch because it shows digits (e.g., 10:05:32) and assume it's very precise.
• Why it's wrong: A typical digital watch (wristwatch) has a least count of 1 second — same as a wall clock. It just displays seconds, but it cannot measure fractions of a second.
• How to avoid: Check the smallest division or smallest digit change the device can reliably measure. For a digital watch, that is 1 s.

Example: A digital watch showing 10:05:32 cannot show 10:05:32.5 — so its least count is 1 s.

Mistake 3: Forgetting the Atomic Clock's Least Count

• What students do: They think an atomic clock is "too complicated" or "not a common clock," so they ignore it.
• Why it's wrong: An atomic clock uses vibrations of atoms (e.g., cesium-133) and is accurate to about 1 part in $10^{13}$ — an uncertainty of order $10^{-13}$ s. This is far smaller than any mechanical or digital watch.
• How to avoid: Memorise the order of magnitude of least counts:
  • Wall clock: 1 s
  • Stopwatch: 0.1 s or 0.01 s
  • Digital watch: 1 s
  • Atomic clock: $\sim 10^{-13}$ s (1 part in $10^{13}$)

Correct answer: (d) An atomic clock — because its least count is the smallest.

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