Through how many radians does the minute hand on a clock rotate in (a) 12 hr and (b) 3 hr?

Key Concepts

Radians and Degrees

Radians are a unit of angular measure used in mathematics, where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. There are 2π radians in a full circle (360 degrees), making radians a natural choice for calculations involving circular motion, such as the rotation of clock hands.

Angular Velocity

Angular velocity refers to the rate of change of angular position of an object, typically measured in radians per unit of time. For a clock, the minute hand completes one full rotation (2π radians) every 60 minutes, which allows us to calculate how many radians it rotates in any given time period by using the formula: radians = (time in minutes) × (angular velocity).

Time Conversion

To solve problems involving the rotation of clock hands, it is essential to convert time into a consistent unit. For instance, converting hours into minutes is necessary since the minute hand's movement is typically measured in minutes. This conversion allows for accurate calculations of the total radians rotated over specified time intervals.