- #1

joeyar

- 53

- 0

**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?

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- Thread starter joeyar
- Start date

- #1

joeyar

- 53

- 0

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

- #2

humanino

- 2,490

- 8

23 times ?

- #3

joeyar

- 53

- 0

Close but no cigar.

- #4

Jimmy Snyder

- 1,095

- 20

It depend upon when you start counting the 12 hours.

- #5

brads77

- 8

- 0

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

- #6

humanino

- 2,490

- 8

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.

- #7

phyzmatix

- 313

- 0

Does it have an hour, minute as well as second hand? Or only hours and minutes?

- #8

brads77

- 8

- 0

OOps--forget previous answer...thinking 24 now...

- #9

brads77

- 8

- 0

I was thinking second hand before

- #10

joeyar

- 53

- 0

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

- #11

humanino

- 2,490

- 8

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

- #12

Jimmy Snyder

- 1,095

- 20

There are others as well.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)

- #13

Mokae

- 20

- 0

Yes, so difficult the problem

- #14

- 12,150

- 169

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

Also ... somebody may want to check if I missed any

There are 22 times where right-angles occur.

Note: times are

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

Last edited:

- #15

DavidWhitbeck

- 351

- 1

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

[tex]\mathbf{x_b}(t) = \sin \omega_b t \mathbf{i} + \cos \omega_b t \mathbf{j}[/tex]

while the position vector of the little hand is given by

[tex]\mathbf{x_l}(t) = \sin \omega_l t \mathbf{i} + \cos \omega_l t \mathbf{j}[/tex]

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

[tex]\mathbf{x_b}(t)\cdot\mathbf{x_l}(t) = 0 \Rightarrow[/tex]

[tex]\cos\omega_b t \cos\omega_l t + \sin \omega_b t \sin \omega_l t = 0 \Rightarrow [/tex]

[tex]\cos (\omega_l - \omega_b)t = 0 \Rightarrow [/tex]

[tex](\omega_l - \omega_b)t_n = \pi n \Rightarrow [/tex]

[tex]t_n = \frac{\pi n}{\omega_l - \omega_b}[/tex]

but

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

So we have

[tex]t_n = \frac{n}{2} \frac{T_b T_l}{T_b - T_l}[/tex]

where

[tex]T_b = 12 h, T_l = 1 h[/tex]

- #16

Kurdt

Staff Emeritus

Science Advisor

Gold Member

- 4,846

- 6

Angular velocity of minute hand: [itex] \omega_m=\frac{2\pi}{3600} [/itex]

Angular velocity of hour hand: [itex] \omega_h = \frac{2\pi}{43200} [/itex]

Goverened by the rotational dynamics equations:

[tex] \theta = \theta_0 + \omega t[/tex]

We want:

[tex] \theta_m - \theta_h = (2n+1) \frac{\pi}{2} [/tex]

and we get:

[tex] t = \frac{(2n+1) \pi}{2(\omega_m - \omega_h)} [/tex]

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.

- #17

DavidWhitbeck

- 351

- 1

- #18

joeyar

- 53

- 0

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 areapproximate.

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.

Last edited:

- #19

phlegmy

- 119

- 0

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?

- #20

Doctoress SD

- 30

- 0

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

- #21

NateTG

Science Advisor

Homework Helper

- 2,452

- 7

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.

- #22

Doctoress SD

- 30

- 0

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

- #23

Kurdt

Staff Emeritus

Science Advisor

Gold Member

- 4,846

- 6

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.

- #24

- 12,150

- 169

https://www.physicsforums.com/showthread.php?t=100534

Color test for e1e1e2:

This is a test.

- #25

Kurdt

Staff Emeritus

Science Advisor

Gold Member

- 4,846

- 6

Comes out as e3e3e3 on photoshop for me. Both should work fine though.

- #26

- 12,150

- 169

(14 x 16) + (3 x 1)

or e3 in hexadecimal.

You're right!

This is color e3e3e3

- #27

Doctoress SD

- 30

- 0

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

- #28

DeaconJohn

- 122

- 0

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 ... somebody may want to check if I missed any

There are 22 times where right-angles occur.

Note: times areapproximate.

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

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.

Last edited:

- #29

- 12,150

- 169

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

I'm counting 22 in my list.

- #30

DeaconJohn

- 122

- 0

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

- #31

- 12,150

- 169

No problem.

Regards,

Mark

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