CHAPTER 12: UNIFORM CIRCULAR MOTION

Uniform Circular Motion describes the movement of an object traveling at constant speed along a circular path. Although the speed is constant, the velocity is continuously changing because the direction changes. This change in velocity means there is acceleration, and therefore a force acting on the object. Understanding circular motion is essential for explaining everything from the orbit of planets to the operation of a centrifuge or the experience of a car turning on a banked road.

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12.1 ANGULAR MEASUREMENT: THE RADIAN

Before studying circular motion, we must understand how to measure angles in radians—the natural unit for rotation.

12.1.1 Definition of the Radian

One radian (rad) is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.

Mathematically, if an arc length s is equal to the radius r, then the angle θ = 1 radian.

12.1.2 Relationship between Radians and Degrees

The circumference of a circle is 2πr. The angle for a full circle (360°) in radians is:

θ = (arc length) / r = (2πr) / r = 2π radians

Therefore:

2π rad = 360°

π rad = 180°

1 rad = 180°/π ≈ 57.3°

1° = π/180 rad ≈ 0.01745 rad

Worked Example (Radian Conversion):

Convert 60° to radians and 1.5 radians to degrees.

Solution: 60° = 60 × (π/180) = π/3 rad ≈ 1.047 rad. 1.5 rad = 1.5 × (180/π) = (1.5 × 180)/3.14 ≈ 270/3.14 ≈ 85.9°.

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12.2 ANGULAR DISPLACEMENT AND ANGULA

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