The hour hand of a clock is 6cm long

General
1

the hour hand of a clock is 6cm long

2

How can we calculate the distance traveled by the tip of the hour hand in a clock?

Answer:

When we talk about the distance traveled by the tip of the hour hand of a clock, we’re referring to the arc length traced as the hand moves. The tip of the hour hand moves in a circular path, and the length of this path for a complete cycle (full rotation) forms the circumference of the circle.

Concept Introduction:

The path traced by the tip of the clock’s hour hand is a circle, where the length of the hour hand is the radius. The complete rotation of the hour hand takes 12 hours to cover the clock’s full face. The key here is to understand the relationship between the radius of the circle and the circumference, which is determined by the formula:

  • Circumference of a Circle: C = 2 \pi r, where ( r ) is the radius.

Calculation Process:

  1. Length of the Hour Hand: 6 cm (this is the radius of the circle formed by the hour hand).

  2. Formula for Circumference:

    The circumference of the path (circle) traced by the hand is calculated using the formula:

    C = 2 \pi r

  3. Plug in the Values:

    • Replace ( r ) with 6 cm in the formula:

      C = 2 \pi \times 6

    • Calculate the Circumference:

      C = 12 \pi \text{ cm}

Distance Traveled in Different Times:

  • Complete 12-hour Cycle: The tip travels the full circumference.

    • Distance: ( 12 \pi ) cm

  • One Hour Rotation: The hour hand completes ( \frac{1}{12} ) of a complete rotation each hour.

    • Distance in one hour: ( 12 \pi \times \frac{1}{12} )
    = \pi \text{ cm}

Final Answer:

  • The distance traveled by the tip of the hour hand in one hour is (\pi) cm.
  • For a complete 12-hour rotation, the distance is ( 12\pi ) cm, equivalent to approximately (37.7 ) cm.

This understanding helps in converting circular time movements into measurable linear distances.