Determine the Exact Time Between 7 and 8 O'clock When the Clock Hands Form a 0° Angle

Have you ever wondered at what specific time between 7 and 8 o'clock the angle between the two hands of a clock is exactly 0°? In this article, we will delve into the mathematics behind this intriguing question and find the precise moment when the clock hands overlap.

The Mathematical Formula

The angle θ between the two hands of a clock can be calculated using the formula:

θ | Mod(60H - 11M/2) | or | Mod(60*7 - 11M/2) | 0

Where:

H is the hour (in this case, 7) M is the minute

By setting the equation to 0, we get:

60*7 - 11M/2 0

Solving for M, we find:

M 420/11 ≈ 38 min 10.9 sec

The Exact Time

Therefore, the hands of the clock are exactly over one another at:

07:38:10.9 or 07:38:11 in HH:MM:SS format

The Movement of the Hands

To understand this calculation better, let's break it down further:

The hour hand moves at a rate of 30^circ/hr. The minute hand moves at a rate of 360^circ/hr. At 8 o'clock, the angle between the hands is 240^circ. The minute and hour hands move at different rates, with the angle between them decreasing at a rate of 360^circ/hr - 30^circ/hr 330^circ/hr. The time it takes for the angle to go from 240^circ to 0^circ is:

T

The Final Answer

Thus, the exact time when the hands overlap between 7 and 8 o'clock is:

08:43:38.overline{18}

This precise calculation is crucial for understanding the intricate dynamics of the clock hands. By applying basic mathematical principles, we can uncover hidden moments in time that are both interesting and educational.