91Ó°ÊÓ

Understanding how to convert radians to degrees is essential when dealing with circular motion or angles. The radian is a standard unit of angular measure used primarily in mathematics. It provides a direct correlation between the length of an arc of a circle and the radius. However, degrees are often used because they are more intuitive for many people. To convert radians to degrees, use the formula:

• Degrees = Radians × (180/Ï€)

This formula is derived from the fact that a full circle is 360 degrees, which is equivalent to 2Ï€ radians. Therefore, 1 radian equals 180/Ï€ degrees. In practical terms, if you know the angle in radians, multiplying by 180/Ï€ converts the measurement into degrees.

Earth's radius plays a crucial role in calculating distances and angles related to geographical locations. The average radius of the Earth is about 4000 miles or approximately 6371 kilometers. It is important to note that this is an approximate value, as the Earth is not a perfect sphere but an oblate spheroid. For calculating angular distance between two points on the Earth's surface, we often use this average radius in calculations. Using Earth's radius helps us gauge the curvature arc and thus determine distances along its surface. For example, when determining the angle between Kampala and Singapore on the equator, we factor in Earth's radius to find the arc length.

Calculating distance is paramount in geography and navigation. The formula used here is part of the principles of circular motion, where an arc distance (S) is related to the radius (r) and central angle (θ) in radians:

• Distance (S) = Radius (r) × Angle (θ)

For instance, in the problem presented, the distance between Kampala and Singapore along Earth's curvature is given as 5000 miles. Knowing Earth's radius is about 4000 miles, we use this formula to find θ: i.e.,

• θ = S / r = 5000 miles / 4000 miles = 1.25 radians

Circular motion is a foundational concept in physics and engineering, describing the movement of an object along the circumference of a circle. A primary aspect of circular motion is angular velocity, which expresses how quickly an object rotates or revolves relative to an axis. Angular velocity ( ω) is particularly useful when examining the motion of an object traveling in a circular path. In this exercise, we find the angular velocity of a plane flying from Kampala to Singapore. The formula used is:

• ω = θ / t

where θ is the angular displacement in radians, and t is the time in hours. Thus, the angular velocity is calculated as:

• ω = 1.25 radians / 9 hours = 0.139 radians per hour

This angular velocity explains how much of the circle's arc the plane covers per hour in its journey.