Displacement and Distance Covered by the Minute Hand of a Wall Clock

To determine the displacement and distance covered by the minute hand of a wall clock from 10 am to 10:30 am, we need to consider the properties of circular motion and basic trigonometry.

Properties of Circular Motion:
1. Radius: The length of the minute hand of a wall clock is given as 10 cm. This length represents the radius of the circular path traced by the minute hand.
2. Angular Velocity: The minute hand completes a full revolution in 60 minutes, or 360 degrees. This gives us an angular velocity of 6 degrees per minute.

Displacement:
The displacement of an object is a vector quantity that represents the change in its position. In the case of circular motion, displacement is the straight-line distance between the initial and final positions of the object.

Calculating Displacement:
1. At 10 am, the minute hand is pointing directly at the 12 o'clock position. This position corresponds to an angle of 0 degrees.
2. At 10:30 am, the minute hand has moved halfway between the 12 o'clock and 1 o'clock positions. This position corresponds to an angle of 15 degrees.
3. The displacement of the minute hand is calculated as the straight-line distance between these two positions, which can be determined using trigonometry.
- The displacement, d, can be calculated using the formula: d = 2πr(Δθ/360), where r is the radius and Δθ is the change in angle.
- Substituting the given values, we have d = 2π(10)(15/360) = π/6 ≈ 0.524 cm.

Distance Covered:
The distance covered by an object is the total length of the path it follows, regardless of the direction. In circular motion, distance is calculated by considering the entire circumference of the circle traced by the object.

Calculating Distance Covered:
1. The circumference of a circle is given by the formula: C = 2πr, where r is the radius.
2. Substituting the given value, we have C = 2π(10) = 20π ≈ 62.83 cm.
3. As the minute hand completes a full revolution in 60 minutes, it covers the circumference of the circular path in this time.
4. In the given time interval from 10 am to 10:30 am, which is 30 minutes, the minute hand covers a fraction of the total circumference.
- The distance covered, D, can be calculated using the formula: D = (Δθ/360) × C, where Δθ is the change in angle and C is the circumference.
- Substituting the given values, we have D = (30/60) × 20π = 10π ≈ 31.42 cm.

Therefore, the displacement of the minute hand from 10 am to 10:30 am is approximately 0.524 cm, and the distance covered by the minute hand in the same time interval is approximately 31.42 cm.