Problem 111 - Algebra and Trigonometry

Question

The angular velocity of a point on Earth is \(\frac{\pi}{12}\) radian per hour. Describe what happens every 24 hours.

Step-by-Step Solution

Answer

A point on Earth makes a complete rotation, or travels \(2\pi\) radians, every 24 hours.

Step 1: Understand What Angular Velocity Is

Angular velocity is a measure of the rate of change of an angle with respect to time. It is given in units of angle per time. Here it's given in radians per hour.

Step 2: Calculate Angular Displacement

To find the total angular displacement after 24 hours, we need to multiply the angular velocity by the time interval. The formula to find the angular displacement is given by: \[ \text{Angular Displacement} = \text{Angular Velocity} \times \text{Time}\] Our given angular velocity is \(\frac{\pi}{12}\) radians per hour and time is 24 hours.

Step 3: Plug In The Values

Substitute the given angular velocity and time into the formula: \[\text{Angular Displacement} = \frac{\pi}{12} \text{ rad/hour} \times 24 \text{ hours}\]

Step 4: Calculate The Final Answer

On multiplying, the 'hour' unit cancels out and the answer we get is \(2\pi\) radians or we can say the point makes a complete rotation about the Earth once in a 24 hours period.