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When we talk about angular speed, we often use the unit 'radians per second'. This can seem confusing at first, but it's simply a way to measure how fast an object is rotating. A radian is a measure of angle based on the radius of a circle. Imagine taking the radius of a circle and bending it along the edge of the circle. The angle subtended by this arc is one radian. There are about 6.28 radians in a full circle, which is the number we refer to using the symbol \(2\pi\).

Back to the exercise in question, the clock's second hand covers this \(2\pi\) radians in 60 seconds. If we want to find out the angular speed in radians per second, we divide the total radians, \(2\pi\), by the time it takes to cover them, which is 60 seconds. This calculation gives us a neat measurement in radians per second, which tells us how quickly any point on the second hand is moving not in terms of distance, but in terms of the angle covered.

The concept of angular displacement is central to understanding motion in a circle. Angular displacement refers to the angle through which a point or line has been rotated in a specified sense about a specified axis. It’s different from linear displacement, which measures how 'far' an object has moved. Angular displacement is measured in radians, degrees, or revolutions, and in the case of our second hand on a clock, we measure it in radians.

In one minute, a second hand experiences an angular displacement of \(2\pi\) radians, or one full rotation. If an object makes more than one full rotation, the angular displacement could be greater than \(2\pi\). It might be helpful to think of angular displacement as how 'far' an object has rotated, in terms of angle, not in the straight-line distance.

The rate of change, in the context of angular motion, is the speed at which the angle changes with respect to time. It is a measure of how quickly or slowly the angular position of an object is changing as it rotates. In our exercise, the rate of change is the angular speed of the clock's second hand. We calculate it by dividing the total angular displacement by the total time taken.

Mathematically, the rate of change of angular displacement with respect to time is known as angular velocity or angular speed. This is symbolized by the Greek letter omega (\(\omega\)). To clarify with numbers: if a second hand moves \(2\pi\) radians in 60 seconds, we divide \(2\pi\) by 60 to get \(\frac{1}{30}\pi\) radians per second. This gives us a clear and precise rate of change, which helps in understanding how fast the second hand is ticking.