Clock Hand Speeds and Relative Motion
The minute hand moves 360 degrees in 60 minutes, which is $6$ degrees per minute ($360/60$).
The hour hand moves 360 degrees in 12 hours (720 minutes), which is $0.5$ degrees per minute ($360/720$).
The relative speed at which the minute hand gains on the hour hand is the difference between their speeds: $6 \text{ deg/min} - 0.5 \text{ deg/min} = 5.5 \text{ deg/min}$. This is equal to $\frac{11}{2}$ degrees per minute.
Calculating Time Between Coincidences
For the hands to coincide again after being together, the minute hand must complete a full circle (360 degrees) relative to the hour hand.
Using the formula: Time = Distance / Speed
The time interval between two consecutive coincidences is:
$ \text{Time} = \frac{360 \text{ degrees}}{5.5 \text{ degrees/minute}} = \frac{360}{\frac{11}{2}} \text{ minutes} $ $ \text{Time} = 360 \times \frac{2}{11} = \frac{720}{11} \text{ minutes} $Converting to Mixed Fraction
To express this time in minutes as a mixed fraction, we divide 720 by 11:
$720 \div 11 = 65$ with a remainder of $5$.
So, the time is $65 \frac{5}{11}$ minutes.
Rohit's Time at Restaurant
Rohit observes the hands coinciding when he arrives around 1 PM and again when he leaves about an hour later. The time elapsed between these two observations is the duration between two consecutive coincidences of the clock hands.
Therefore, Rohit spent $65 \frac{5}{11}$ minutes at the restaurant.
Important Questions from Clocks
- On a given day, how many times will the second-hand and the minute-hand of a clock cross each other during the clock time 12:05:00 hours to 12:55:00 hours?
- A worker noticed that the hour hand on the factory clock had moved by 225 degrees during her stay at the factory. For how long did she stay in the factory?
- In a 12-hour clock that runs correctly, how many times do the second, minute,and hour hands of the clock coincide, in a 12-hour duration from 3 PM in a dayto 3 AM the next day?
- It is quarter past three in your watch. The angle between the hour hand and the minute hand is ______
- The minute-hand and second-hand of a clock cross each other __________ times between 09:15:00 AM and 09:45:00 AM on a day.
