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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.

Circular Motion 

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Kinematics of Uniform Circular Motion

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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

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An object moving with a constant speed in a curved path changes its direction of velocity continuously. Because of the velocity changes, acceleration is produced. The direction of velocity of an object which exhibits the uniform circular motion is along the tangent to the circular path. The changes in the direction of velocity is towards the center of the circular path. So,the centripetal acceleration is defined as the acceleration of an object which is in circular motion and the direction of the acceleration is towards the center of the circular path. Centripetal The meaning of centripetal is 'towards the center' . The mathematical expression for centripetal acceleration is given as,

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

Centripetal Force

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The centripetal acceleration is produced by a force directed toward the center of circular motion termed as the centripetal force. If Fc represents centripetal force and m be the mass of an object in uniform circular motion, then the mathematical equation for centripetal force is,

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

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