In Exercises 87–90, find the absolute value of the radian measure of the angle that the second hand of a clock moves through in the given time.55 seconds

Verified step by step guidance

1. Understand that the second hand of a clock completes a full circle, which is 360 degrees or \(2\pi\) radians, in 60 seconds.
2. Determine the fraction of the full circle that the second hand moves through in 55 seconds by calculating \(\frac{55}{60}\).
3. Multiply the fraction \(\frac{55}{60}\) by \(2\pi\) to find the radian measure of the angle moved by the second hand in 55 seconds.
4. Simplify the expression \(\frac{55}{60} \times 2\pi\) to find the radian measure.
5. Take the absolute value of the radian measure obtained to ensure the result is non-negative.

Key Concepts

Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. One radian is the angle formed when the arc length is equal to the radius of the circle. In a full circle, there are 2π radians, which corresponds to 360 degrees. Understanding radian measure is essential for converting time into angular movement, especially in circular motion problems.

Angular Velocity

Angular velocity refers to the rate at which an object rotates around a central point, typically measured in radians per second. For a clock, the second hand completes one full revolution (2π radians) in 60 seconds. This concept is crucial for determining how far the second hand moves in a given time frame, allowing us to calculate the angle in radians that corresponds to the time elapsed.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. In the context of angles, it ensures that we consider only the magnitude of the angle moved, without regard to whether it is positive or negative. This is particularly important when dealing with angles in circular motion, as it provides a clear representation of the extent of movement.