Radians

• Radian is a unit of angular measurement that represents the angle subtended at the center of a circle where the arc length equals the radius.

  Key points:

  • Natural angular measurement in mathematics and physics
  • 1 radian is the angle where arc length = radius
  • Fundamental to describing circular motion
• magine a circle. If you take the radius of that circle and wrap it around the circle's edge, the angle created at the center is one radian.

  Think of it like measuring an angle by using the circle's own radius as a measuring tape. This makes radians a very natural way to measure angles.
• A full circle's circumference is$2 \pi r$, which is about 6.28 times the radius length. So when you wrap the radius around the circle's edge completely, you create an angle of$2\pi$radians (approximately 6.28 radians).

  
  

  A full circle is2πradians. A quarter of a circle would be2π/4=2π​radians, which is approximately 1.57 radians.
• • A full circle can be measured in two ways
  • In degrees, a full circle is 360°
  • In radians, a full circle is$2\pi$

  So if you want to convert:

  • Half a circle = 180° =$\pi$radians
  • Quarter of a circle = 90° =$\frac{\pi}{2}$radians
• Calculating radians involves understanding the relationship between arc length, radius, and the angle.

  The formula is:$\text{Radians} =$$\frac{\text{Arc Length}}{\text{Radius}}$

  Example:

  • If an arc length is 10 cm
  • And the radius is 5 cm
  • Radians = 10/5 = 2 radians
• Imagine you're measuring an angle by using the circle's radius as a measuring stick:

  • If the arc length is exactly the same as the radius, that's 1 radian
  • If the arc length is twice the radius, that's 2 radians
  • If the arc length is half the radius, that's 0.5 radians

  It's like measuring the angle by seeing how many "radius lengths" fit along the curved part of the circle.
• Radians are crucial in describing circular motion in physics:

  • Used to calculate angular velocity
  • Describe rotational motion
  • Help calculate linear velocity

  Key applications:

  • Describing rotation of wheels
  • Pendulum motion
  • Planetary orbits
  • Mechanical engineering
• Think of radians like a special "ruler" for measuring how things spin:

  • When a bicycle wheel turns, radians tell you exactly how much it rotates
  • A full wheel spin =$2\pi$radians
  • Half a wheel spin =$\pi$radians

  Imagine a point on the wheel's edge moving in a circle. Radians help you track exactly how far and fast it moves.