Circular Motion

The revolution of moon around the Earth and the revolution of an artificial satellite in a circular orbit round the Earth are examples of circular motion.

Consider a particle moving with constant speed along the circumference of a circle of radius R in the anticlockwise direction. The time taken by the particle to go round the circle once is called the time period of the particle and is denoted by the letter T.

The number of revolutions made by the particle in one second is defined as the frequency of revolution of the particle. The frequency is denoted by the symbol n. If n revolutions are completed in one second, then one revolution will be completed in 1/n second. But the time taken to complete one revolution is the time period T.

During one complete revolution, the radius vector sweeps out an angle of 2p radians in time T, the periodic time.

Therefore, the angular velocity or angular frequency

= Angular displacement/time taken

Angular velocity or angular frequency expressed by

Now, with reference to the figure below.

Let O, the centre of the circle, be chosen as the origin in the plane. Let the moving particle be at the points P and Q of the circle at time t and (t + Dt) respectively. Let the respective position vectors be(t) and(t + Dt). Let the arc PQ subtend an angle Dq at the centre of the circle

Then, the angular velocity,

In order to calculate the velocity vectorof the particle at P, let us divide the displacement vector by the time interval Dt and ultimately take the limit as Dt approaches zero.

Moreover, since the sum of the angles of a triangle is 180^o or D radius,

As Dt becomes smaller and smaller,

The velocity (t) of the particle at P is along the tangent to the circle at P. Similarly, the vector(t + Dt) of the particle at Q is along the tangent to the circle at Q. In both cases, the magnitude of the velocity is wR(constant speed). In order to calculate the change in velocity in time Dt, let us draw both the velocity vectors (t) and(t + Dt) from the same point P.

the triangle formed by the vectors and is an isosceles triangle.

 

 

where f is the angle which D (change in velocity in time Dt) makes with . When Dt approaches zero, the acceleration is in the direction of Its magnitude is w²R, from equation (i).

Therefore, we conclude that when a particle moves with a constant speed in a circle, the velocity is always tangential and has an acceleration, which is directed radially inwards. Both velocities and accelerations have constant magnitudes but changing directions.