Minute Hand Tip Movement Calculation

This solution details how to calculate the distance covered by the tip of a watch's minute hand over a specific period.

Determining Given Values

We are given the following information:

• Length of the minute hand, which acts as the radius (r) of the circular path: $r = 1.5\text{ cm}$
• Time interval: $t = 40\text{ minutes}$
• Value of pi to use: $\pi = 3.14$

Calculating the Angle Covered

The minute hand completes a full circle ($360^\circ$) in $60$ minutes.

• Angle swept per minute = $\frac{360^\circ}{60} = 6^\circ$ per minute.
• Total angle swept in $40$ minutes = $40 \times 6^\circ = 240^\circ$.

Converting Angle to Radians

For calculations involving arc length, the angle must be in radians.

• To convert degrees to radians, multiply by $\frac{\pi}{180^\circ}$.
• Angle in radians ($\theta$) = $240^\circ \times \frac{\pi}{180^\circ} = \frac{4}{3}\pi$ radians.

Calculating the Distance (Arc Length)

The distance the tip moves is the arc length (s) along the circle.

The formula for arc length is $s = r\theta$, where $r$ is the radius and $\theta$ is the angle in radians.

• Substitute the known values: $s = (1.5\text{ cm}) \times (\frac{4}{3}\pi)$
• Simplify the expression: $s = \frac{3}{2} \times \frac{4}{3} \pi \text{ cm} = 2\pi \text{ cm}$

Final Distance Calculation

Substitute the given value of $\pi = 3.14$ into the simplified expression:

• $s = 2 \times 3.14 \text{ cm}$
• $s = 6.28 \text{ cm}$

Therefore, the tip of the minute hand moves $6.28\text{ cm}$ in $40\text{ minutes}$.