At 6 o'clock the hands of a clock are exactly in line At what time are they again in line?

Answer 1

Seven wrong answers so far. Apply a method similar to that described by Inquirer (the 7th answer) to find all 48 times this happens every day.

The other correct answer is 'none', since my digital clock doesn't have hands.

Answer 2

You are asking for the times when the hands will be straight again. That means when the minute hand has advanced one full revolution farther than the hour hand.

I find this is easiest in terms of relative velocity. The minute hand goes around 12 times in 12 hours. The hour hand is once every 12 hours. So the minute hand goes 11 revolutions farther than the hour hand in 12 hours, or 11/12 revolution per hour.

To go one more revolution than the hour hand takes:

1 revolution /(11/12 revolution per hour) = 12/11 hours.

So every 1 1/11 hours the hands line up. 1/11 hour is 5 minutes and 27.2727 seconds. (the 27 repeats)

So start with 6 o'clock and keep adding 1:05:27.2727

You get:

6:00:00
7:05:27.2727
8:10:54.5454
9:16:21.8182
10:21:49.0909
11:27:16.3636
12:32:43.6364
1:38:10.9091
2:43:38.1818
3:49:05:4545
4:54:32.7273
6:00:00

Answer 3

6:32:43.64
Note that "in line" means either 0 deg or 180 deg. Between 6:30
and 6:35 the minute hand will cross the hour hand. It can be found that it is in fact 6:32:43.64 (rounded to 100th of a second).

If you want to know how to calculate this: Let m be the minute part of the time at which the alignment occurs. The corresponding hour angle position is not independent. It is constrained by the minute part of the time. The angle A made by both the hands to 12 O' clock reference line is given by:

A = 6xm - 180

The same angle A is made by the hour hand at this time:
A/30 = m/60

Eliminating A with the above two eqns, this makes

m = 180x2/11

Hence the time of alignment is 6 : 32 : 43.64

Answer 4

The minute hand strikes 6 ranges a minute. The hour hand strikes a million/2 degree a minute. because of the fact the minute and hour arms are 30 ranges aside (a million/12 of the circle) on the start, it takes 30/(6-a million/2) minutes after a million:00 for them to examine up. just to 30/5.5 or 60/11. subsequently, the arms would be precisely on perfect of one yet another 5 5/11 minutes after one basically the time would be a million:05:40 5.40 5, rounded to the closest hundredth of a 2nd. in easy terms for the checklist, at a million:06 the arms would be 3 ranges aside.

Answer 5

I suppose that by "in line" you mean forming an angle of 180 degrees or Pi rads.
Assuming that the hands are facing the same direction at 12 o'clock we can express their displacements as

theta = theta_initial + omega * t,

where theta_initial may be regarded as zero for both hands.

where theta is the angle and omega the angular velocity.
We know that omega = 2*Pi/T, where Pi=3.14159 and T the period of rotation.
For the small hand we have T2=1 hr and for the big hand we have T1 = 12 hr.
So we need to solve the equation
dTheta = Theta2 - Theta1 ->
dTheta = 2*Pi*t/T2 - 2*Pi*t/T1 ->
dTheta = 2*Pi*t*(1/T2-1/T1) ->
dTheta = 2*Pi*t*(T1-T2)/(T1*T2) ->
t = dTheta*T1*T2/[2*Pi*(T1-T2)]
Since dTheta = Pi the above formula may be simplified as

t = T1*T2/[2*(T1-T2)]
which results in t = 0.55 hr (approximate value)
So after 12 noon the two hands are exactly in line at 12:32.7

We can apply the same technique for the question given.
At 6 o'clock we have Theta1_Initial = Pi and Theta2_Initial=0

So
Theta1=Pi+2*Pi*t/T1
Theta2=2*Pi*t/T2
dTheta = Theta2 - Theta1 ->
dTheta = 2*Pi*t/T2 - (Pi+2*Pi*t/T1) ->
dTheta = 2*Pi*t*(1/T2-1/T1) -Pi ->
Pi = 2*Pi*t*(T1-T2)/(T1*T2)-Pi ->
1 = 2*t*(12-1)/(12*1)-1 ->
1 = 22*t/12-1 ->
2 = 22*t/12 ->
t = 24/22 hr ->
t = 12/11 hr

So the answer is 12/11 hr after six o'clock or approximately at 7:05.46

Answer 6

My guess is before 7:05.
between 6 and 7:05 the hours and shifts slowly towards the 7. It's not a continuous move, so it moves bit by bit.
Now, at the same time, the minutes hand moves bit by bit as well.
If the increments coincide (say, if 2 steps of the minute hand equal one step of the hour hand, for example) then they might be at opposite direction before 7:05.

So bottom line: 7:05 for sure, and depending on the structure of the clock, possibly before that.

Answer 7

7 0 clock

Answer 8

At 7:07.

Answer 9

Quater past 3

Answer 10

they would be in line at:
25 minutes to 1
20 minutes to 2
15 minutes to 3
10 minutes to 4
5 minutes to 5
6 o'clock
5 minutes past 7
10 minutes past 8
15 minutes past 9
20 minutes past 10
25 minutes past 11
12 o'clock