Hour & Minute Hand: Straight Lines & Overlaps

[Mk]

How many times per day (00:00-24:00) do the hour and minute hand form a straight line together? Would this be the same for when they overlap? Say, the minute hand goes pi/1 hr and the hour hand goes (pi/12)/1 hr. Then what?

 

[Werg22]

How many times per day (00:00-24:00) do the hour and minute hand form a straight line together? Would this be the same for when they overlap? Say, the minute hand goes pi/1 hr and the hour hand goes (pi/12)/1 hr. Then what?

Construct a function and find it's solutions according to what you are looking for. As a hint, both hand start on 12 at 0:00. Now with the frequency of each hand, can you find how does the angle between them change with time?

I've put a "solution" here. I covered it though, it's only there if you really have no reason to try solving the problem by yourself (maybe a homework?). You can highlight it to see it, but I recommend to try to come up with the solution yourself.


Assume we are dealing with a unit circle. The velocity of a hand, actually equal to it’s angular velocity because we’re dealing with a unit circle, is equal to 2π/T, where T is the period of the hand. For the minute hand we have v = 2π and for the hour hand we have v2 = 2π/60. Now note that the distance along the circle covered by a hand is equal to v*t, where t is the time elapsed in minute. Moreover, the distance between the two hands is given by the difference of the respective covered distances. Since this is on a unit circle, this distance is equal to the angle that separates the two hands. We end up with something like this as a function θ(t) = 2π*t - 2π/60*t or θ(t) = 59*π*t/30. Now, it’s obvious that if θ is to be a multiple of π, t = k*30/59, where k is any integer number. However we have to be careful. If θ is a multiple of 2π, the hands are certainly not opposite, they are overlapping. This is equivalent to saying that the value k has to be odd. Let’s solve k*30/59 = 1440 (there are 1440 minutes in day). We have k = 2352. So there are 2352 instance of the day where is a multiple of π. Now, since of have the requirement of k being odd, there 2352/2 = 1176 solutions of interest.

 

[tacman]

The hour and minute hand on a clock form a straight line twice a day, once at 12:00 and once at 12:30. This is because the hour hand moves in increments of 30 degrees and the minute hand moves in increments of 6 degrees. Therefore, at 12:00, the hour hand is at 0 degrees and the minute hand is at 0 degrees, forming a straight line. At 12:30, the hour hand is at 180 degrees and the minute hand is at 90 degrees, forming another straight line.

When the hour and minute hand overlap, it happens 22 times per day. This is because the hour hand moves in increments of 30 degrees and the minute hand moves continuously. So, every time the minute hand makes a full rotation, it will overlap with the hour hand 22 times.

In the scenario given, where the minute hand moves at a rate of pi/1 hour and the hour hand moves at a rate of (pi/12)/1 hour, the hour and minute hand will form a straight line every 2 hours. This is because the hour hand will move 30 degrees every hour and the minute hand will move 6 degrees every hour, which will result in a straight line every 2 hours.

When it comes to overlapping, the number of times they overlap will still be 22 times per day. This is because the rate at which they move does not affect the number of times they overlap. It only affects the frequency at which they form a straight line together.