6.1.1 Circular motion

Circular Motion Basics

Angular Speed (ω)

Angular speed ($ω$) represents the angle an object moves through per unit time. It can be calculated using either:

$$\omega = \frac{v}{r} \quad \text{or} \quad \omega = \frac{2\pi}{T} = 2\pi f$$

Where:

• $v$ = linear speed,
• $r$ = radius of the circular path,
• $T$ = time period for one full revolution,
• $f$ = frequency (number of revolutions per second).

Radians and Angular Measurements

Angles in circular motion are often measured in radians. One radian is the angle formed when the arc length is equal to the radius of the circle. The full circle is $2\pi$ radians. Conversions:

• Degrees to radians: multiply by $\frac{\pi}{180}$
• Radians to degrees: multiply by $\frac{180}{\pi}$

Formulas for Circular Motion

1. Centripetal Acceleration ($a$):

$$a = \frac{v^2}{r} = \omega^2 r$$

This formula shows that acceleration depends on the speed and radius of the circular path.

2. Centripetal Force (F): Using Newton's Second Law $F = ma$ the formula for centripetal force becomes:

$$F = \frac{mv^2}{r} = m \omega^2 r$$

Where:

• $m$ = mass of the object,
• $v$ = speed of the object,
• $r$ = radius of the circle,
• $\omega$ = angular speed.