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# Motion in a Circle

Motion in a circle or circular motion is found in many situations in our daily life, such as a roller coaster traveling near the top or bottom of its track, a car traveling around a turn, the Earth orbiting the Sun and a centrifuge. An object with uniform circular motion travels in a circle with a constant speed. If a golf ball tied to a string is whipping around in circles. The ball is traveling at a uniform speed as it follows a circular path, so we can say that it is moving in uniform circular motion. The given figure illustrate the circular motion.

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## Kinematics of Uniform Circular Motion

Circular motion is defined as motion of a body in a curved path due to a perpendicular force, at constant speed.

Angular displacement: Angular displacement (θ) of a body defined as the directed angle with respect to the center of the arc travel.

i.e, S = rθ
$\theta$ = $\frac{S}{r}$

Where S = arc length (m)
$\theta$ = angular displacement (radians or rad)
r = radius of arc (m)

• SI unit of angular displacement is the radian (rad). It is a vector quantity, perpendicular to the direction of motion.
• One radian is the angle subtended by an arc length equal to the radius of the arc.
• For one complete revolution, the angular displacement = 2π rad
• From above, since 360° = 2π, then 1 rad = $\frac{360^{\circ}}{2\pi }$ or 1° = $\frac{\pi }{180^{\circ}}$

Angular velocity: Angular velocity ($\omega$) of a body in circular motion is defined as the rate of change in angular displacement (θ) with respect to time (t).

$\omega$ = $\frac{\mathrm{d} \theta }{\mathrm{d} t}$  = 2πf

Where $\omega$ = angular velocity (rad s-1)
f = frequency of oscillation (Hz)

• SI unit of angular velocity is rad s-1. It is a vector quantity perpendicular and out of plane to the direction of motion.
• Period (T) is defined as the time taken for one complete oscillation.
• Frequency (f) is defined as the number of complete oscillations per unit time.

## Centripetal Acceleration

a = $\frac{v^{2}}{r}$
Fc = ma = m$\frac{v^{2}}{r}$