Tru Physics

Uniform Circular Motion

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Tru Physics
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Introduction

Uniform circular motion refers to the motion of an object traveling in a circular path at a constant speed. While the speed remains constant, the direction of the velocity vector changes continuously, resulting in a non-zero acceleration.

Velocity in Uniform Circular Motion

The velocity of an object in uniform circular motion is given by:

v = \dfrac{2 \pi r}{T}

where:

  • v is the velocity,
  • r is the radius of the circular path,
  • T is the period of the motion (the time for one complete revolution).

Centripetal Acceleration

Even though the speed of the object remains constant, the object is accelerating because the direction of its velocity vector is continuously changing. This acceleration, known as centripetal acceleration, is directed towards the center of the circular path. The centripetal acceleration is given by:

a_c = \dfrac{v^2}{r}

where:

  • a_c is the centripetal acceleration,
  • v is the velocity,
  • r is the radius of the circular path.

Alternatively, centripetal acceleration can be expressed in terms of the period T:

a_c = \dfrac{4 \pi^2 r}{T^2}

Centripetal Force

The net force causing the centripetal acceleration is known as the centripetal force. By Newton’s second law, the centripetal force is given by:

F_c = m \cdot a_c

where:

  • F_c is the centripetal force,
  • m is the mass of the object,
  • a_c is the centripetal acceleration.

Substituting the expression for centripetal acceleration, we can also write the centripetal force as:

F_c = \dfrac{mv^2}{r}

or, in terms of the period T:

F_c = \dfrac{4 \pi^2 m r}{T^2}

Frequency and Angular Velocity

The frequency f of the motion, which is the reciprocal of the period, is given by:

f = \dfrac{1}{T}

The angular velocity \omega, which is the rate of change of the angle subtended at the center of the circle, is given by:

\omega = \dfrac{2 \pi}{T} = 2 \pi f

Conclusion

Uniform circular motion is a fundamental concept in physics. It has applications in a wide range of areas, including planetary motion, the operation of centrifuges, and the design of roads and tracks. Understanding the principles of uniform circular motion is essential to understanding these and many other physical phenomena.

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