How many times do the clock hands form a 120 - degree angle in 12 hours?

Hey there! As a clock hands supplier, I've been thinking about all sorts of clock - related questions. One interesting one that popped into my head is: How many times do the clock hands form a 120 - degree angle in 12 hours? Let's dig into this fun little math problem.

First off, we need to understand how the hour hand and the minute hand move. The minute hand makes a full rotation (360 degrees) in 60 minutes, so its angular velocity is 360/60 = 6 degrees per minute. The hour hand makes a full rotation in 12 hours (or 720 minutes), so its angular velocity is 360/720 = 0.5 degrees per minute.

Let (t) be the number of minutes passed after 12:00. The angle of the minute hand (\theta_m=6t) degrees, and the angle of the hour hand (\theta_h = 0.5t) degrees.

We want to find when the absolute difference between the angles of the two hands is 120 degrees. So we have two cases:

Case 1: (\theta_m-\theta_h = 120), which means (6t - 0.5t=120). Combining like terms, we get (5.5t = 120), and then (t=\frac{120}{5.5}=\frac{240}{11}\approx21.82) minutes.

Case 2: (\theta_h-\theta_m = 120), which means (0.5t - 6t=120), or (- 5.5t=120), and (t =-\frac{240}{11}). Since time (t>0), we ignore this negative solution in our normal time - counting context. But we also need to consider the full - cycle situation.

The two hands will form a 120 - degree angle multiple times in 12 hours. In one hour, the relative angular velocity between the minute hand and the hour hand is (6 - 0.5=5.5) degrees per minute.

A full - cycle difference between the two hands is 360 degrees. To form a 120 - degree angle again after the first occurrence, the relative movement of the two hands needs to cover either 120 degrees (to get to the next 120 - degree situation in the same "direction") or (360 - 120 = 240) degrees (to get to the 120 - degree situation in the opposite "direction").

Let's calculate the number of times they form a 120 - degree angle in 12 hours.

In 12 hours (720 minutes), the number of times they form a 120 - degree angle:

We know that the relative angular displacement between the two hands to form a 120 - degree angle (either increasing or decreasing the angle between them) is based on the relative angular velocity of 5.5 degrees per minute.

The number of times they form a 120 - degree angle (n):

The total relative angular displacement in 12 hours is considered. If we consider the two possible relative displacements (120 degrees and 240 degrees between consecutive 120 - degree situations), we find that they form a 120 - degree angle 22 times in 12 hours.

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In conclusion, the clock hands form a 120 - degree angle 22 times in 12 hours. And we're here to provide you with the best clock hands in the market.

References:

• Basic knowledge of circular motion and angular velocity in physics textbooks.
• Mathematical analysis of clock - hand problems from various math education resources.