Circular Motion
Now we shall discuss another example of two-dimensional motion that is motion of a particle on a circular path. This type of motion is called circular motion.
Consider a particle P is moving on circle of radius r on X – Y plane with origin O as centre.
The position of the particle at a given instant may be described by angle q, called angular position of the particle, measured in radian. As the particle moves on the path, its angular position (q) changes. The rate of change of angular position is called angular velocity (w) measured in radian per second.
The rate of change of angular velocity is called angular acceleration, measured in rad/s2. Thus, the angular acceleration is
RELATION BETWEEN DIFFERENT PARAMETERS OF CIRCULAR MOTION
ü It is easy to derive the equations of rotational kinematics for the case of constant angular acceleration with fixed axis of rotation. These equations are of the same form as those for one-dimensional translational motion.
w = w0 +αt …(i)
Φ = Φ0 + w0t + αt2/2 …(ii)
w2 = w02 + 2α (Φ- Φ0) …(iii)
Φ= Φ0 + (w0 +w)/(2t) …(iv)
Here,Φ is the initial angle and w0 is the initial angular speed.
Illustrations
1. (a) What is the angular velocity of the minute and hour hands of a clock?
(b) Suppose the clock starts malfunctioning at 7 AM which decelerates the minute hand at the rate of 4p radians/day. How much time would the clock loose by 7 AM next day?
Sol. Angular speed of
minute hand : wmh = 2Π rad/hr
= 48Π rad/day = (Π/1800) rad/sec
hour hand : whh = (Π/6) rad/hr
= 4Π rad/day = (Π/21600) rad/sec
(b) Assume at t = 0, Φ0 = 0, when the clock begins to malfunction. Use equation (ii) to get the angle covered by the minute hand in one day.
Hence the minute hand complete 23 revolutions, so the clock losses 1 hour.