define the radian and express angular displacement in radians

Kinematics of Uniform Circular Motion

What is a Radian?

A radian is a unit of angular measure that comes from the geometry of a circle. Imagine a circle with radius r. If you take a piece of string equal to that radius and lay it along the circumference, the angle subtended by that arc is one radian. Mathematically, one radian is defined as the angle θ for which the arc length s equals the radius: $$s = r\,θ \quad\text{and}\quad θ = 1\;\text{rad} \;\text{when}\; s = r.$$ So, 1 radian ≈ 57.3°.

Why Use Radians?

• Radians give a direct relationship between arc length and angle: s = rθ.
• They simplify many formulas in physics, e.g., angular velocity ω = dθ/dt.
• They are the natural unit in calculus because the derivative of sin θ is cos θ only when θ is in radians.

Expressing Angular Displacement in Radians

Angular displacement (Δθ) is the change in the angle of a rotating object. If an object moves along a circular path, the arc length travelled (Δs) is related to Δθ by: $$Δs = r\,Δθ.$$ Rearranging gives: $$Δθ = \frac{Δs}{r}.$$ So, to find the angular displacement in radians, divide the distance travelled along the circle by the radius.

Practical Example 🚀

1. Take a 2 m radius wheel.
2. The wheel rolls 6 m along the ground.
3. Angular displacement: $$Δθ = \frac{6\,\text{m}}{2\,\text{m}} = 3\;\text{rad}.$$
4. In degrees, that’s about $$3 \times 57.3° \approx 171.9°.$$

Quick Conversion Table 🔄

Key Takeaway 🎯

When you’re dealing with circular motion, always think in radians: they keep the maths neat and let you directly link distance, radius, and angle with the simple formula Δθ = Δs / r.