A Time puzzle to solve...

[FenceFurniture]

Looking at the face of a regular analogue clock, at what time between 1pm and 2pm is the minute hand precisely positioned over the hour hand?

No cheating, and you must supply a proof of your answer!

[419]

Can't calculate it precisely (I never went past Form 2 maths and wasn't any good at that), but I reckon it's around 1.05 pm.

Reasoning is that the hour hand has to move through 30 degrees (1/12th of 360 degrees) on a 12 hour clock face for each hour while the minute hand has to move through 360 degrees each hour, so 360/30 =12. An hour has 60 minutes so divided by 12 = 5, thus 1 pm + 5 = 1.05pm.

[rwbuild]

1:06:06 pm because the minute hand advances 1 minute exactly for every 1 full rotation of the second hand plus 6 sec for each complete rotation of the second hand

[419]

1:06:06 pm because the minute hand advances 1 minute exactly for every 1 full rotation of the second hand plus 6 sec for each complete rotation of the second hand

I might be missing something here (and quite likely am), but wouldn't that result in 1:05:06?

Admittedly, I'm wedded to my brilliant deduction that it's 1:05 as the base. :)

[ian]

Looking at the face of a regular analogue clock, at what time between 1pm and 2pm is the minute hand precisely positioned over the hour hand?

No cheating, and you must supply a proof of your answer!
It largely depends on the mechanism controlling the clock hands.

Is your analogue clock a Mondial one as used by the Swiss railways?
In those the minute hand jumps forward 6 degrees as the second hand reaches the 12 o'clock position, but the hour hand creeps forward at a regular rate of 30 degrees per hour. Given that, the hour hand is never precisely above the hour hand between 1 and 2 o'clock.
So the question is moot.

[mic-d]

I did it this way

533500

if the answers not clear at the bottom it is 1:05.45pm
and I see I forgot to write in 'and the hour hand moves at 0.5º/min' when I did my neat rightup

[rwbuild]

I might be missing something here (and quite likely am), but wouldn't that result in 1:05:06?

Admittedly, I'm wedded to my brilliant deduction that it's 1:05 as the base. :)

No because at 1Pm the minute hand is at 12 and hour hand at 1 so when the minute hand moves to exactly the 1 position the hour hand has also moved proportionately past the 1

[FenceFurniture]

All I can say is none of the answers are right so far. The time will obviously have to be between 1:05 and 1:10 though.

And it's not a Mondial clock, because that would take the fun out out of it.

[FenceFurniture]

Extracting the correct parts of answers:

the hour hand has to move through 30 degrees (1/12th of 360 degrees) on a 12 hour clock face for each hour while the minute hand has to move through 360 degrees each hour, so 360/30 =12.Correct. But if the minute hand (M) has already moved though 30°, how far has the hour hand (H) moved as well? Obviously the M catches up to the H before 1:10. Perhaps Trial & Error might get you closer to the accurate (as opposed to "around") time? Having said that, there is a very precise formula that can be derived.

[mic-d]

All I can say is none of the answers are right so far. The time will obviously have to be between 1:05 and 1:10 though.

And it's not a Mondial clock, because that would take the fun out out of it.

Can you point out the error in my maths please FF? if you need it converted to seconds 1:05.45pm would be 1:05:27pm

[FenceFurniture]

Can you point out the error in my maths please FF? if you need it converted to seconds 1:05.45pm would be 1:05:27pmAh! I read your answer as 45 seconds past 1:05. So in that case your answer is indeed correct Mick! It's actually closer to 1:05:28 because the seconds are 27.73.

As follows:

Every second the M hand moves 0.1°, and the H hand moves 1/12 of that, or 0.083333 repeater°
So if H starts at 30° and s is the seconds until M = H then
30 + 0.083333 * s = 0 + s * 0.1
Multiply both by 10 ► s = 300 + s*0.83333

or ► s = 300 + s/12
Multiply both by 12 ► 12s = 300*12 + s
so ► 11s = 3600
so ► s = 3600/11
= 327.73 seconds later
or 1:00 plus 327.73 seconds which is 1:05 and 27.73 seconds.

[mic-d]

Ah! I read your answer as 45 seconds past 1:05. So in that case your answer is indeed correct Mick! It's actually closer to 1:05:28 because the seconds are 27.73.

As follows:

Every second the M hand moves 0.1°, and the H hand moves 1/12 of that, or 0.083333 repeater°
So if H starts at 30° and s is the seconds until M = H then
30 + 0.083333 * s = 0 + s * 0.1
Multiply both by 10 ► s = 300 + s*0.83333

or ► s = 300 + s/12
Multiply both by 12 ► 12s = 300*12 + s
so ► 11s = 3600
so ► s = 3600/11
= 327.73 seconds later
or 1:00 plus 327.73 seconds which is 1:05 and 27.73 seconds.



If I account for my rounding error, it is actually 1:05:27.2727272727...pm. You need to check your maths 3600/11

Cheers
M

[FenceFurniture]

If I account for my rounding error, it is actually 1:05:27.2727272727...pm. You need to check your maths 3600/11

Cheers
MYeah, a misread of the calculator.

[ian]

By brute force -- and assuming the clock mechanism allows for continuous movement of the minute hand, and 1 second (= 6 degrees) jumps by the second hand ...

At 1:05
the minute hand has moved to 30 degrees around the clock face
the hour hand has moved 30/12 (= 2.5 degrees) from its starting position at 30 degrees, so it's total movement around the clock face is 32.5 degrees -- the minute hand is clearly behind the hour hand.

at 1:06
the minute hand has moved a further 6 degrees, so it is 36 degrees around the clock face,
the hour hand has moved to 33 degrees around the face -- clearly the minute hand is ahead of the hour hand.

at 1:05:30
the minute hand will be at a position 33 degrees around the clock face,
the hour hand will have moved to a position [30 + (30/60 x 5.5)] = 32.75 degrees around the clock face, marginally behind the minute hand, but can you determine the difference in the angle?

at 1:05:24
the minute hand will be at a position 32.7 degrees around the clock face,
the hour hand will be positioned at [30 + (30/60 x 5.4)] = 32.7 degrees


DONE

.
.
.
.

I hope

[FenceFurniture]

at 1:05:24
the minute hand will be at a position 32.7 degrees around the clock face,
the hour hand will be positioned at [30 + (30/60 x 5.4)] = 32.7 degrees


DONE

.

...not quiteM moves 360° in 60 minutes or 3600 seconds, so 0.1° per second.
H moves at 1/12 that velocity
After 324 seconds (at 1:05:24):
M will have moved 32.4° from 0°, and be in position 32.4° but
H will have moved 1/12 of that (2.7°) from 30° position to be at 32.7°.

That means that M is still 0.3° (or 3 seconds) behind H. That means the time has to be at least 1:05:27
However, following on from that, in those additional 3 seconds H has moved another (0.1 * 3)/12° or 0.025°,
hence a further ~¼ of a second (actually 0.272727 to 6 decimal places, but let's round it to 0.3) has to be added to the time for Swiss precision.

[FenceFurniture]

We know that both H and M hands are constantly moving, which means that at 1:05:27.2727 seconds the centre of M moves over the centre of H, but it is only for an instant. We have also established that M moves at 12x the velocity of H, but neither are ever still.


Is your analogue clock a Mondial one as used by the Swiss railways?
In those the minute hand jumps forward 6 degrees as the second hand reaches the 12 o'clock position, but the hour hand creeps forward at a regular rate of 30 degrees per hour. Given that, the hour hand is never precisely above the hour hand between 1 and 2 o'clock.
So the question is moot.This is not true. My opening statement of this post means that the question is indeed not moot for a Mondial clock. In this case, M moves at an obviously much faster speed than in a regular clock, but in both cases the M hand is moving across H – just at a different speed. That means there is indeed an instant in time when the hands are perfectly centred together – it is just briefer by the ratio of the speeds of the two different M hands..

If I knew the velocity of M (when it moves, or even how long it takes to move) in a Mondial clock of choice I believe I could work out that precise instant of time to any choice of decimal places. :D

[Tonyz]

Ive been working 2 days straight serving cruise ship people, Iam tired and as long as my eyes are closed before 1AM comes I couldnt care :D

[Skew ChiDAMN!!]

in those additional 3 seconds H has moved another (0.1 * 3)/12° or 0.025°,
hence a further ~¼ of a second (actually 0.272727 to 6 decimal places, but let's round it to 0.3) has to be added to the time for Swiss precision.

So, to be even more precise we can say it occurs at 5 5/11 minutes past 1? :D

[taz01]

Slightly different way, using simultaneous equations

Using T to represent the time in minutes the rotation of the hour hand on the clock face is

H = T / (60 x 12) + 1/12

1/12 representing the starting position of 1pm

Minute hand is just M = T/60. One revolution every 60mins.

When aligned M = H, so T/60 = T/ (60×12) + 1/12

This solves to T = 60/11
Giving
M = 1/11

This is 5.4545... minutes in decimal or 5min 27 and 27/99 seconds

[mic-d]

Slightly different way, using simultaneous equations

Using T to represent the time in minutes the rotation of the hour hand on the clock face is

H = T / (60 x 12) + 1/12

1/12 representing the starting position of 1pm

Minute hand is just M = T/60. One revolution every 60mins.

When aligned M = H, so T/60 = T/ (60×12) + 1/12

This solves to T = 60/11
Giving
M = 1/11

This is 5.4545... minutes in decimal or 5min 27 and 27/99 seconds


That's basically the way I did it

[GraemeCook]

On my antique mantle clock, the hands would be at 6:30 ..... because the clock is buggered and the hands are always at 6:30.

Another project.

[ian]

If I knew the velocity of M (when it moves, or even how long it takes to move) in a Mondial clock of choice I believe I could work out that precise instant of time to any choice of decimal places. :DHi Brett

to assist.
the minute hand on a Swiss Railway Mondaine clock (I shouldn't trust my very faulty memory when it comes to spelling) -- jumps forward 6 degrees over 1/16th of a second when the second hand reaches the 12 o'clock position. The second hand jumps forward 6 degrees every second.

BTW, the vibration frequency of the quartz crystal found in most analogue clocks is 32.768 kHz -- thousand cycles per second. So if you are wishing to be precise with the time the minute and hour hands are precisely aligned, you will need to factor in the quartz crystal vibration frequency.

[FenceFurniture]

Later.

[419]

BTW, the vibration frequency of the quartz crystal found in most analogue clocks is 32.768 kHz -- thousand cycles per second. So if you are wishing to be precise with the time the minute and hour hands are precisely aligned, you will need to factor in the quartz crystal vibration frequency.

The original question doesn't limit the type of analog clock to a quartz movement, unless one interprets 'regular' in the question to mean as regular as a quartz movement as distinct from a 'regular' 12 number clock face.

If the latter, then it includes clocks with mechanical movements, the accuracy of which depend upon the quality of manufacture of the gears and springs etc. If so, isn't it impossible to calculate the alignment of the hands on a given mechanical clock without knowing further details of its deviation from a theoretical standard equivalent to a quartz movement as the mechanical hands may be ahead of or after the hands on a quartz movement?

[419]

On my antique mantle clock, the hands would be at 6:30 ..... because the clock is buggered and the hands are always at 6:30.

Confirms the old comment that even a broken clock is correct twice a day.

[FenceFurniture]

If so, isn't it impossible to calculate the alignment of the hands on a given mechanical clock without knowing further details of ....................................................................................................................Ummmm, I actually don't think I care.



It's just a bit of fun....no need to join in bogging it down in pedantry.

[419]

Ummmm, I actually don't think I care.



It's just a bit of fun....no need to join in bogging it down in pedantry.

Sorry, I must have missed the bits in this thread that weren't pedantic.

[FenceFurniture]

Well you didn't miss this, and it's most definitely not pedantic.
I reckon it's around 1.05 pm.

Like I say, it's just supposed to be a bit of fun.

[419]

Like I say, it's just supposed to be a bit of fun.

Fair enough.

For a bit of fun, could you explain how your not pedantic calculations and assessments of contributors' various answers apply equally to predicting the position of the hands on a mechanical analogue clock of unknown manufacture and unknown accuracy?

[FenceFurniture]

Fair enough.

For a bit of fun, could you explain how your not pedantic calculations and assessments of contributors' various answers apply equally to predicting the position of the hands on a mechanical analogue clock of unknown manufacture and unknown accuracy?No. Because I couldn't be fagged. :;

[419]

No. Because I couldn't be fagged. :;

Fair enough.

I'm sure you must be exhausted after this sort of precision answering your own general question about the position of the hands on an unspecified analog clock with an unspecified movement and commenting on the accuracy, deficiencies or otherwise of those who answered your problem.


Every second the M hand moves 0.1°, and the H hand moves 1/12 of that, or 0.083333 repeater°
So if H starts at 30° and s is the seconds until M = H then
30 + 0.083333 * s = 0 + s * 0.1
Multiply both by 10 ► s = 300 + s*0.83333

or ► s = 300 + s/12
Multiply both by 12 ► 12s = 300*12 + s
so ► 11s = 3600
so ► s = 3600/11
= 327.73 seconds later
or 1:00 plus 327.73 seconds which is 1:05 and 27.73 seconds.


Or could it be that it is impossible to identify the position of the hands on a non-quartz movement analog clock except by observation of a given clock because the machining and tensions and inter-operation of the various parts introduces unknown and incalculable variables which were not contemplated by the problem and formulas you posed?

No need to respond if you actually don't think you care or couldn't be fagged.

Just a bit of fun.

[mic-d]

Thank you FF for posting these maths related quizzes and taking the time to provide answers. It's got the grey matter working on logic and recall of high school maths which I'm grateful for. I hope you keep posting them when you have time.

And no need to begin with a long preamble about how this is an idealised situation and provide 25 caveats, it's implicit that it's the ideal case. Could you imagine an exam question written like that! Or how Einstein would have got around to explaining his theories of relativity if people piped up with 'wait, but is the train steam, electric or diesel, or my god, is it diesel-electric, and is there a nice restaurant car on it that serves a good BLT sandwich?' or 'is the rocket ship solid or liquid fuelled, does it have Netflix to make the time go faster?'

I see cheekiness in this post (the quartz clock) but also nastiness in the form of a Straw Man argument (the mechanics of a clock) designed to impugn the strength of your question, it's a compensation mechanism by those who couldn't or wouldn't solve it, much in the way a flash sports car is compensation for ED. :q:q

[yvan]

All I can say is none of the answers are right so far. The time will obviously have to be between 1:05 and 1:10 though.

And it's not a Mondial clock, because that would take the fun out out of it.

All I can say is Thank God for digital timepieces !!!
I haven't got the time to figure out at what precise time one hand is over the over, at whatever hour...
As the piece's movement is continuous, I can only estimate that yours is a trick question!
For a milli-second there I thought I had the answer but the bloody thing moved again....

All I can say is we'll get to Xmas on the 25th.

Merry Christmas
Yvan

[Boringgeoff]

Reminds me of the time a policeman pulled me up and accused me of failing to stop at a stop sign. I explained to him the physics of an insect hitting your windscreen head on, in the split second between when the insect is flying forwards until it starts going backwards there is a moment when it is stationary, at this precise moment the vehicle is also stationary. Needless to say, the policeman was not impressed by my brilliant scientific knowledge and the ticket was duly written.

Cheers,
Geoff.

[ian]

The original question doesn't limit the type of analog clock to a quartz movement, unless one interprets 'regular' in the question to mean as regular as a quartz movement as distinct from a 'regular' 12 number clock face.

If the latter, then it includes clocks with mechanical movements, the accuracy of which depend upon the quality of manufacture of the gears and springs etc.
"regular" in horological terms refers to the escapement mechanism invented by John Harrison around 300 years ago. The escapement mechanism allowed the spring spring tension to be released in a controlled manner so that the hands on a regulated clock always move at a constant speed regardless of the spring tension.


Horology is the study and measurement of time and the art of making of clocks and watches.

[FenceFurniture]

Yes, I guess that makes John Harrison the father of "proper" navigation. You'd think they'd have the decency to call at Harrology wouldn't you? :D

[ian]

Yes, I guess that makes John Harrison the father of "proper" navigation. You'd think they'd have the decency to call at Harrology wouldn't you?
Horology has been the term used for the study of time and clocks for a very very long time.

The book Longitude by Dava Sobel tells the story of how Harrison solved "the greatest scientific problem of Harrison's time"
I highly recommend the book.

[FenceFurniture]

I am not going to consider:

the time it takes M to accelerate and decelerate in its 6° travel in 1/16 second – it's unknown anyway
anything to do with quartz
wear and tear, age, quality, country of manufacture, the weight of any dust on the hands, wind direction, gravitational pull, La Niña effect or any other such ridiculous, pedantic notion to satisfy the 81st Prime Number


Assumptions:

M starts moving at the very end of each 60 second period, rather than at 59 and 15/16 seconds (i.e. a 1/16 sec before the minute has finished).
M and H are both symmetrical in their width shape, and are straight


Givens:

We know that at 1:05 plus 1/16 seconds, M is at the 30° position, and that it only moves every 60 seconds.
We know that at 1:06, M is still at the 30° position, but 1/16 second later it is at 36°.
We know that at 1:05, H is at the 32.5° position, and that it advances at a rate of 0.5° per minute or 1/120° per second.


Thus, at 1:06:00, H has moved to 33.0°, and 1/16 seconds later it has moved 1/120/16° or 0.00052° (rounded) and will be at 33.00052°

So when will M be at 33.00052°?
A 1/16 second is 0.0625 seconds so it has to be a little over half of that time to move a little over half the distance (3.00052° of 6° travel)
So 3.00052 / 6 * 0.0625 = 0.03125542534 seconds
At 1:06 plus 0.03126 seconds to the nearest 1/10,000 sec is when the centre of M passes across the centre of H, or more realistically phrased as 1:06 plus 1/32 seconds (in other words plus half the 1/16 second it takes to click forward).

[ian]

I am not going to consider:

the time it takes M to accelerate and decelerate in its 6° travel in 1/16 second – it's unknown anyway
anything to do with quartz
wear and tear, age, quality, country of manufacture, the weight of any dust on the hands, wind direction, gravitational pull, La Niña effect or any other such ridiculous, pedantic notion to satisfy the 81st Prime Number


Doses that mean that you will allow the coriolis force to apply ??


:D


more [possibly] when I have the time

[FenceFurniture]

I see cheekiness in this post (the quartz clock)Agreed.


but also nastiness in the form of a Straw Man argument (the mechanics of a clock) designed to impugn the strength of your question, it's a compensation mechanism by those who couldn't or wouldn't solve it, much in the way a flash sports car is compensation for ED. :q:qAlso agreed. Real fun stuff, and completely uncalled for.


For clarity (definitions from internet search):
accurate adjective (especially of information, measurements, or predictions) correct in all details, exact

So giving rise to "as accurate as possible" if it can't be exact or precise, and not to be confused with

pedantic adjective excessively concerned with minor details or rules; over-scrupulous. Pointing out minor errors, correcting people who make small mistakes, and bragging about their own knowledge and expertise

[FenceFurniture]

Doses that mean that you will allow the coriolis force to apply ??


:D


more [possibly] when I have the time:bartmoon:

[GraemeCook]

... The book Longitude by Dava Sobel tells the story of how Harrison solved "the greatest scientific problem of Harrison's time"
I highly recommend the book.

I totally endorse Ian's recommendation. It was a major research project. Before that, coasts were discovered by running into them ... literally.

A major part of Cook's Endeavour voyage was trialling a chronometer. Did it work in practice as well as in theory? It did.

The French were also equally involved in similar research. On d'Entrecasteaux's voyages some 20 years later he carried 14 chronometers (# from memory). The Brits wanted to know if chronometers actually worked, the French wanted to know which was best! Science was advancing rapidly.

[rwbuild]

.....at the third stroke the time will be..ah bugger.....at the fourth stroke the time will be..ah bugger.....at the fifth.... bugger it, lets have a beer instead

[FenceFurniture]

.....at the third stroke the time will be..ah bugger.....at the fourth stroke the time will be..ah bugger.....at the fifth.... bugger it, lets have a beer insteadThat guy, with his BBC accent, is long gone! So is the service. :D

[yvan]

.....at the third stroke the time will be..ah bugger.....at the fourth stroke the time will be..ah bugger.....at the fifth.... bugger it, lets have a beer instead

Quite right! A few sherberts later and time doesn't matter any more!
I'll have to ask FF is this qualifies as an answer to his original trick question:rolleyes: