Understanding Clock Hands and Angular Movement

Clock problems often involve calculating the angular distance covered by the hands. The minute hand is one of the fastest moving hands on a standard clock face, completing a full circle in one hour.

To solve this problem, we need to determine the angular speed of the minute hand and then calculate the total angle covered for the given duration.

Angular Speed of the Minute Hand

The minute hand completes one full revolution, which is 360 degrees, in 60 minutes.

Therefore, the angular speed of the minute hand can be calculated as:

Angular speed = $\frac{\text{Total degrees in a circle}}{\text{Total time for one revolution}}$

Angular speed = $\frac{360^{\circ}}{60 \text{ minutes}}$

Angular speed = $6^{\circ}\text{ per minute}$

This means that for every minute that passes, the minute hand moves 6 degrees.

Calculating Total Time in Minutes

The question states that the minute hand covers a distance equivalent to a duration of 1 hour and 10 minutes.

• 1 hour = 60 minutes
• Given time = 1 hour and 10 minutes

Total time in minutes = 60 minutes + 10 minutes = 70 minutes.

Calculating Angular Distance Covered

Now that we have the total time in minutes and the angular speed of the minute hand, we can calculate the total angular distance covered.

Angular distance = Angular speed $\times$ Total time

Angular distance = $6^{\circ}\text{/minute} \times 70 \text{ minutes}$

Angular distance = $420^{\circ}$

So, when the minute hand covers a duration of 1 hour and 10 minutes, it moves through an angle of 420 degrees.

Summary of Calculation

Comparing with Options

Let's compare our calculated angular distance with the given options:

• (a) 400°
• (b) 410°
• (c) 420°
• (d) 430°

Our calculated angular distance of 420° matches option (c).

Revision Table: Clock Hand Angles

Additional Information: Understanding Angular Distance

Angular distance refers to the angle through which an object rotates or moves along a circular path. In the context of a clock, as the hands move, they trace out angles from their starting position.

The concept of angular speed is crucial for solving clock problems. It is the rate at which the angle changes over time.

• A full circle is 360 degrees.
• When a hand moves from 12 back to 12, it has covered 360 degrees.
• 1 hour on a clock face corresponds to $360^{\circ}/12 = 30^{\circ}$ for the hour hand.
• 1 minute mark on a clock face corresponds to $360^{\circ}/60 = 6^{\circ}$ for the minute hand or second hand.

Calculating angular distance is a direct application of the relationship between speed, time, and distance, adapted for rotational motion.