Question

Through how many radians does the hour hand on a clock rotate in (a)
24 hrand (b) 4 hr?

Answer 1

The hour hand on a clock rotates through 4π radians in 24 hours, and through 2π/3 radians in 4 hours, based on the proportion between the time and the total number of hours for a full 2π radian rotation.

Step-by-step explanation:

The amount of rotation for the hour hand on a clock can be calculated considering a full circle equals 2π radians, which corresponds to 12 hours on a clock. Thus, we can determine the rotation in radians for any amount of time by setting up a proportional relationship.

Calculating Rotation in Radians

(a) For a 24-hour period, the hour hand completes two full rotations, which means it rotates through 2 x 2π radians. Therefore:

• 24 hours corresponds to 4π radians (since 1 full rotation is 2π radians).

(b) For a 4-hour period, we calculate the fraction of the total 12-hour rotation, which would be ⅔ or 1/3 of 2π radians. Therefore:

• 4 hours corresponds to (1/3) x 2π = 2π/3 radians.