1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Clock hands!

  1. Feb 23, 2007 #1


    User Avatar

    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?
  2. jcsd
  3. Feb 23, 2007 #2
    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.
    Last edited: Feb 23, 2007
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook