How many times are the hands of a clock at right angle in a day?

The simple mathematical approach can be taken as follows to solve “How many times are the hands of a clock at a right angle in a day?”: In a 12 hour period, the minute hand makes 12 turns while the hour hand makes one turn.

If you switch to a rotating coordinate system where the hour handstands, the minute hand only makes 11 turns and 22 times at right angles to the hour hand.

In 24 hours a day, you get 2 × 22 = 44, times the hands make right angle.

How many times are the hands of a clock at right angle in a day? — How many times are the hands of a clock at right angle in a day

Number of Times When the hands of a clock are at the right angle in a day (hours hand and minute hand) in ante meridiem and post meridiem:

Ante Meridiem	Post Meridiem
12:16 AM	12:16 PM
12:49 AM	12:49 PM
1:21 AM	1:21 PM
1:54 AM	1:54 PM
2:27 AM	2:27 PM
3:00 AM	3:00 PM
3:32 AM	3:32 PM
4:05 AM	4:05 PM
4:38 AM	4:38 PM
5:11 AM	5:11 PM
5:43 AM	5:43 PM
6:16 AM	6:16 PM
6:49 AM	6:49 PM
7:22 AM	7:22 PM
7:54 AM	7:54 PM
8:27 AM	8:27 PM
9:00 AM	9:00 PM
9:33 AM	9:33 PM
10:05 AM	10:05 PM
10:38 AM	10:38 PM
11:11 AM	11:11 PM
11:44 AM	11:44 PM

Here you can see in detail where the hands of the clock are at the right angle. There are two times when the hour’s hand appears in the post and ante meridiem appears only one time literally and i.e 2 AM, 2 PM period and 8 AM, 8 PM period. This happens when the 4th quarter completes.

Why 48 is wrong?

It happens twice every 12 hours, once every 15 after and once a quarter, sometimes two for twenty-four hours = 48. But this decision has one drawback, namely that the clock is progressing, the minute’s needle positioned 90 degrees later in the hour.

Let us check the wrong approach most of us did for the first time.

clock problem false answer — wrong approach

Here we took both 8:55 AM and 9:00 AM both of which will never be possible until someone discovers a time machine. And we also took 2:55 and 3:00 together, where we did our fourth mistake, hence we got two more than the exact answer. If we subtract our errors, then we got the correct answer that is 44 (=48-4). Hope this will make you understand easily.

For more logical maths nerds let’s take an example, a 3:XX alignment at 90 degrees occurs at 3:33 (not 3:30) because the hour hand on the dial has passed 90 degrees, so the minute hand passes after “6”. In fact, at exactly 9:00, the minute hand will expire from the hour hand, for example, a 7:xx sync occurs at 07:54, but no 8:xx sync – it actually happens at 9:00. So we only get 11 90 degrees orientations in 12 hours. And it can double quarter to quarter because of that.

So 2 x 11 for 12 hours = 22 for twelve hours and 44 for 24 hours.

Answer: 44 times the hands of a clock at right angle in a day.

Physical Solution to times when the hands of a clock are at a right angle in a 24 hours?

We know that: 1440 minutes in 24 hours.

The angular speed of the hour hand is 0.5° per minute.
The angular speed of the minute hand is 6° per minute.
In the clockwise direction, the angular speed of the minute hand is 5.5° per minute.
The angle is zero when the time is 12:00.

=> 90 ° / 5.5 gives 16.3636363636 minutes
=> 1440 / 16.36 is 88 (rounded up)
This is for 4 quadrants.
Since we only need to look at 2 quadrants.
So the answer is 88/2 = 44

So, here we discuss two different ways to solve a problem “How many times are the hands of a clock at right angle in a day?” in two approaches: mathematical and applied physics.