# In theory, this should be easy

joeyar
In theory, this should be easy...

... During the space of 12 hours, how many times will the big hand of a clock form a right angle with the little hand?

humanino
23 times ?

joeyar
Close but no cigar.

Jimmy Snyder
It depend upon when you start counting the 12 hours.

59x2--each click on the clock can have a right hand right angle and left handed one...

humanino
It depend upon when you start counting the 12 hours.

Yes. If you start counting at 3:00 or 9:00 for 12:00+epsilon hours, with epsilon positive as small as you want, it seems it makes 23 times. If epsilon is negative, that probably makes only 22. phyzmatix
Does it have an hour, minute as well as second hand? Or only hours and minutes?

I was thinking second hand before

joeyar
Completely ignore the seconds hand, we are only considering the minutes and hours hand. For argument's sake, start at the current time as I post this here in my timezone, which is 1.05.

But it seems to me that humanino has probably got it. :)

humanino
But it seems to me that humanino has probably got it. :)
Well, I got the first (3:00) yesterday before going to bed, and I stupidily was happy about myself, so I forgot the other one (9:00) Jimmy Snyder
Well, I got the first (3:00) yesterday before going to bed, and I stupidily was happy about myself, so I forgot the other one (9:00) There are others as well.

Mokae
Yes, so difficult the problem

Staff Emeritus
Homework Helper
I have used a "brute force" method, simply writing down the (approximate) times at which right-angles occur. Highlight the text below for the solution.

Also ... somebody may want to check if I missed any There are 22 times where right-angles occur.

Note: times are approximate.

12:15
12:50
1:20
1:55
2:25
3:00
3:30
4:05
4:35
5:10
5:40
6:15
6:45
7:20
7:55
8:25
9:00
9:30
10:05
10:40
11:10
11:45

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DavidWhitbeck
There is an analytic way to approach the problem--

The position vector of the big hand is given by (where t=0 is noon)

$$\mathbf{x_b}(t) = \sin \omega_b t \mathbf{i} + \cos \omega_b t \mathbf{j}$$

while the position vector of the little hand is given by

$$\mathbf{x_l}(t) = \sin \omega_l t \mathbf{i} + \cos \omega_l t \mathbf{j}$$

and you require that they are perpendicular to each other so that

$$\mathbf{x_b}(t)\cdot\mathbf{x_l}(t) = 0 \Rightarrow$$

$$\cos\omega_b t \cos\omega_l t + \sin \omega_b t \sin \omega_l t = 0 \Rightarrow$$

$$\cos (\omega_l - \omega_b)t = 0 \Rightarrow$$

$$(\omega_l - \omega_b)t_n = \pi n \Rightarrow$$

$$t_n = \frac{\pi n}{\omega_l - \omega_b}$$

but

$$\omega_b = 2\pi/T_b, \omega_l = 2\pi/T_l$$ where the T's are the periods.

So we have

$$t_n = \frac{n}{2} \frac{T_b T_l}{T_b - T_l}$$
where
$$T_b = 12 h, T_l = 1 h$$

Staff Emeritus
Gold Member
Well its fairly easy to work out. angular velocity of each hand is simple.

Angular velocity of minute hand: $\omega_m=\frac{2\pi}{3600}$
Angular velocity of hour hand: $\omega_h = \frac{2\pi}{43200}$

Goverened by the rotational dynamics equations:

$$\theta = \theta_0 + \omega t$$

We want:

$$\theta_m - \theta_h = (2n+1) \frac{\pi}{2}$$

and we get:

$$t = \frac{(2n+1) \pi}{2(\omega_m - \omega_h)}$$

First couple of answers are (assuming you start from 12:00):

12:16:21.82
12:49:05.45
..etc

EDIT: I was wondering why my tex had come out as vectors for a second.

DavidWhitbeck
Oh yeah woops I made a rookie mistake-- should have been (2n+1)/2 \pi as Kurt has it and not \pi n. Duh!

joeyar
I have used a "brute force" method, simply writing down the (approximate) times at which right-angles occur. Highlight the text below for the solution.

Also ... somebody may want to check if I missed any There are 22 times where right-angles occur.

Note: times are approximate.

12:15
12:50
1:20
1:55
2:25
3:00
3:30
4:05
4:35
5:10
5:40
6:15
6:45
7:20
7:55
8:25
9:00
9:30
10:05
10:40
11:10
11:45

You have the right number. Well done.

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phlegmy
wow someone brought maths into it.
i'd say for the first hour
when the minute hand is 15mins past the hour hand
for the last hour
when the minute hand is 15mins befor the hour hand

for the 10 hours in between
at 15mins before and after the hour hand

10*2 + 2*1 =22?

Doctoress SD
Are we counting every second, minute or hour? LOL, J/K.

Homework Helper
This is pretty easy stuff:

The big hand makes 11 laps relative to the small hand. There are 2 times per lap that it forms a right angle, so the number is 2*11=22.

Doctoress SD
What's with making the text invisible? It doesn't even work, the text is still visible.

Staff Emeritus
Gold Member
To prevent spoilers to those who want to solve it themselves is the purpose. You have to use the correct colour however which is e3e3e3 I believe.

Edit: I have seen other forums with some sort of system that blacks out the text of spoilers which you have to mouse over to reveal. I wonder if that could be applied to this forum.

Staff Emeritus
Homework Helper
The correct color is now e1e1e2 (as of Dec. 2007)

Color test for e1e1e2:

This is a test.

Staff Emeritus
Gold Member
Comes out as e3e3e3 on photoshop for me. Both should work fine though.

Staff Emeritus
Homework Helper
I checked and am getting 227 for each color channel (R, G, B). In base 16 that's ... hmmm ...
(14 x 16) + (3 x 1)

You're right!

This is color e3e3e3

Doctoress SD
In another board I went on they use the code ...

DeaconJohn

I have used a "brute force" method, simply writing down the (approximate) times at which right-angles occur. Highlight the text below for the solution.

Also ... somebody may want to check if I missed any There are 22 times where right-angles occur.

Note: times are approximate.

12:15
12:50
1:20
1:55
2:25
3:00
3:30
4:05
4:35
5:10
5:40
6:15
6:45
7:20
7:55
8:25
9:00
9:30
10:05
10:40
11:10
11:45
Yep, you missed one. The right angle also occurs somewhere around 2:55. On the other hand, you got the right answer, so, you probably had 2:55 in your original calculations. In your "hidden" list, there are only 21 numbers.

Also, I don't think that it depends on the time you start counting for the following reason.

Suppose that you picked a random starting time on Friday afternoon and counted how many times the right angle occurred for the next 12 hours.

Now suppose you want to test if it makes a difference when you start. So, you repeat the same experiment starting Saturday afternoon, but this time, you start a little bit later. Just enough later so that you miss the first right angle that you counted during the Friday afternoon experiment. If anything, you will count one right angle less. But, after you have counted for 12 hours you will find that the last right angle you count will be the one that you missed Saturday afternoon because you delayed your starting time.

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Staff Emeritus
Homework Helper

Yep, you missed one. The right angle also occurs somewhere around 2:55.
Umm, that particular right angle occurs exactly at 3:00, which was in my list.

On the other hand, you got the right answer, so, you probably had 2:55 in your original calculations. In your "hidden" list, there are only 21 numbers.

I'm counting 22 in my list.

DeaconJohn

Umm, that particular right angle occurs exactly at 3:00, which was in my list.

I'm counting 22 in my list.

You are so right on both counts. Thanks for letting me know.

DJ

Staff Emeritus