Calculate Clock Hands Perpendicular Time Post 12:00
The question asks for the first time after 12:00:00 when the hour hand and minute hand of a clock are perpendicular (form a $90^\circ$ angle).
Clock Hand Speeds and Relative Motion
- The minute hand moves $360^\circ$ in 60 minutes, which is $6^\circ$ per minute.
- The hour hand moves $360^\circ$ in 12 hours (720 minutes), which is $0.5^\circ$ per minute.
- The relative speed of the minute hand with respect to the hour hand is $6^\circ/\text{min} - 0.5^\circ/\text{min} = 5.5^\circ/\text{min}$.
Finding the Perpendicular Time
At 12:00:00, both hands are aligned at the 12. We need to find the time $t$ (in minutes) after 12:00:00 when the angle between them is $90^\circ$.
The relative angle covered by the minute hand with respect to the hour hand is given by:
$ \Delta \theta = \text{Relative Speed} \times t $For the hands to be perpendicular, the relative angle must be $90^\circ$.
$ 90^\circ = 5.5^\circ/\text{min} \times t $Solving for $t$:
$ t = \frac{90}{5.5} \text{ minutes} $ $ t = \frac{900}{55} \text{ minutes} $ $ t = \frac{180}{11} \text{ minutes} $Converting Time to HH:MM:SS
Convert the fraction of minutes into minutes and seconds:
$ t = 16 \frac{4}{11} \text{ minutes} $This means 16 full minutes past 12:00.
Now, convert the fractional part ($\frac{4}{11}$ minutes) into seconds:
$ \text{Seconds} = \frac{4}{11} \times 60 = \frac{240}{11} \text{ seconds} $ $ \text{Seconds} \approx 21.818... \text{ seconds} $So, the exact time is 12 hours, 16 minutes, and $21 \frac{9}{11}$ seconds ($12:16:21 \frac{9}{11}$).
Conclusion
Comparing this with the options, the first time after 12:00:00 when the hands are perpendicular is approximately 12:16:21.
Therefore, option A is the correct answer.
Important Questions from Clock and Calendar (Notes)
- From the given options, at what angle are the hands of a clock inclined at 10 minutes to 2 (Smaller angle) ?
- If $1^{st}$ January, 2001 was a Monday, what was the day on $26^{th}$ January, 2003 ?
- Today is Friday. Before $43$ days, it was?
- How many times starting at 1:00 pm would the minute and hour hands of a clock make an angle of $40^\circ$ with each other in the next 6 hours?
- A man starts his journey at 0100 Hrs local time to reach another country at 0900 Hrs local time on the same date. He starts a return journey on the same night at 2100 Hrs local time to his original place, taking the same time to travel back. If the time zone of his country of visit lags by 10 hours, the duration for which the man was away from his place is
