Question

Through how many radians does the minute hand of a clock rotate in \(15 \mathrm{~min}\) ?

Short answer

The minute hand rotates \(\frac{\pi}{2}\) radians in 15 minutes.

Step-by-step solution

Understanding the Problem

To find out how many radians the minute hand of a clock rotates in 15 minutes, we first need to know how many radians it rotates in one full cycle or 60 minutes.

Complete Cycle of the Minute Hand

The minute hand completes one full cycle on the clock, which is a full circle of 360 degrees, in 60 minutes. In terms of radians, a full circle is equivalent to \(2\pi\) radians.

Radian per Minute Calculation

Since the minute hand moves \(2\pi\) radians in 60 minutes, each minute represents \(\frac{2\pi}{60}\) radians. Simplifying gives \(\frac{\pi}{30}\) radians per minute.

Calculate Radians for 15 Minutes

To find how many radians the minute hand rotates in 15 minutes, we multiply the radians per minute by 15: \(15 \times \frac{\pi}{30} = \frac{15\pi}{30} = \frac{\pi}{2}\) radians.

Key concepts

Minute Hand Movement

The movement of the minute hand on a clock is a fascinatingly consistent occurrence. Every clock face is divided into 60 equal parts, each representing one minute. The minute hand of a clock completes a full circle, that is, a 360-degree rotation, every 60 minutes. This means that for each minute, the minute hand moves one-sixtieth of that full circle.

• One full cycle of the minute hand is equivalent to the entire circumference of the circle.
• Each minute the hand shifts represents a small, yet precise portion of this entire circle.
• This consistent and regular motion ensures the passage of time is equally measured.

To understand the angle through which it moves, we use radians, a unit of measure more suited for circular paths.

Radian Measure

Radians provide a measurement alternative to degrees for angles, particularly useful in mathematical calculations involving circles. A full rotation around a circle is equivalent to 360 degrees or, in radians, to \(2\pi\) radians. This is useful because it aligns more naturally with mathematical formulas and equations that involve cycles and periodic functions.

To convert minutes of movement into radians:

• We first need to know how many radians the hand moves in one full circular motion, which is \(2\pi\) radians.
• From there, we can calculate how far the minute hand moves in each minute.
• By dividing the full rotation radians by 60, we understand that each minute moves the hand \(\frac{\pi}{30}\) radians.

Using this conversion, we can easily deduce movements over different periods, such as the 15-minute span relevant to the original exercise.

Clock Physics

Clock physics involves understanding how the parts of a clock move and operate to keep time. Specifically, the focus is often on how hands rotate around the clock face. The minute hand's incremental and consistent movement is a practical example of rotational motion. This movement isn't chaotic but is derived from mechanical precision, ensuring consistency in timekeeping.

Key aspects to consider include:

• Regularity: Completing a full circle, or 360 degrees, every hour maintains uniform time intervals.
• Radians for Calculation: These allow us to express angles in terms of the arc length and radius, streamlining calculations in physical systems.
• Simplicity in Motion: Despite the complexity of its construction, the outcome is one of simple, understandable rotational physics.

Understanding clock physics not only helps us to tell time but also provides foundational insights into rotational motion and the practical use of radians in everyday life.