Understanding Minute Hand Angular Movement on a Clock

A clock's hands move in a predictable way, covering specific angular distances over time. The question asks about the angular distance covered by the minute hand when it moves for 1 hour and 30 minutes.

Minute Hand Speed

The minute hand completes a full circle, which is 360 degrees, in 60 minutes (or 1 hour). To find out how many degrees the minute hand covers in one minute, we can use the following calculation:

Angular distance covered by minute hand in 60 minutes = \(360^{\circ}\)

Angular distance covered by minute hand in 1 minute = \(\frac{360^{\circ}}{60 \text{ minutes}}\)

Angular distance covered by minute hand in 1 minute = \(6^{\circ} \text{ per minute}\)

So, the minute hand moves \(6^{\circ}\) every minute.

Calculating Angular Distance for 1 Hour 30 Minutes

The time given is 1 hour and 30 minutes. First, we need to convert this entire time into minutes.

1 hour = 60 minutes

Total time = 1 hour + 30 minutes

Total time in minutes = 60 minutes + 30 minutes = 90 minutes

Now, we know the minute hand covers \(6^{\circ}\) per minute. To find the total angular distance covered in 90 minutes, we multiply the degrees per minute by the total number of minutes.

Total angular distance covered = Angular distance per minute \(\times\) Total minutes

Total angular distance covered = \(6^{\circ}/\text{minute} \times 90 \text{ minutes}\)

Total angular distance covered = \(540^{\circ}\)

Therefore, when the minute hand of a clock covers a distance of 1 hour 30 minutes, the angular distance covered by it is \(540^{\circ}\).

Summary of Minute Hand Movement

Here is a quick summary of the minute hand's movement:

Revision Table: Clock Hand Speeds

Additional Information: Clock Hand Angles

Understanding how different hands on a clock move is important for solving problems involving angles. Each hand has a constant speed.

• The Minute Hand is the fastest hand usually considered in these problems (apart from the second hand). Its speed is \(6^{\circ}\) per minute. This means for every minute that passes, the minute hand sweeps through an angle of 6 degrees.
• The Hour Hand is much slower. It moves \(360^{\circ}\) in 12 hours. This works out to \(30^{\circ}\) per hour, or \(0.5^{\circ}\) per minute. When the minute hand moves from 12 to 12 (60 minutes), the hour hand only moves from one number to the next (e.g., from 3 to 4), which is \(30^{\circ}\).
• The Second Hand, if present, moves \(360^{\circ}\) in 60 seconds (1 minute). Its speed is \(6^{\circ}\) per second, or \(360^{\circ}\) per minute.

Problems often ask for the angle between two hands at a specific time. These problems require calculating the position of each hand relative to the 12 o'clock mark (usually considered \(0^{\circ}\)) and then finding the difference between their positions. However, this specific question only asks about the total angular distance covered by the minute hand itself, which is a simpler calculation based purely on time elapsed and the minute hand's speed.