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Circular Motion
The Radian  
When discussing the motion of objects which are moving in circular (or nearly circular) paths, we usually consider angles measured in radians rather than degrees.  
   
   
As we can see here, the size of an angle can be measured by comparing the length of the arc, s, with the radius, r of the circle.  
s is directly proportional to r so the ratio s/r is a constant, for a given angle.  
   
The angle θ in radians is defined as follows  
 
So, if we take an arc of length equal to the radius of the circle, we have s/r = 1rad.  
Also, if s is the full circumference of the circle, s/r = 2π which means that 2πrad corresponds to 360 (so 1rad is about 57).  
It should be clear that the radian is, in a certain sense, a more "fundamental" unit of measurement of angle than the degree.  
The degree is used mainly for arithmetical simplicity... 360 divides easily by lots of numbers.  
   
Angular Displacement  
If we consider a small body to be moving round the circle from A to B we say that it has experienced an angular displacement of θ radians.  
The relation between the (linear) distance moved, s, of the body and the angular displacement, θ is obviously given by  
 
Note that, if the angle is very small, then s is very nearly equal to the magnitude of the linear displacement (ie going from A to B along the arc of the circle is very nearly the same distance as going from A to B in a straight line.)  
   
Angular Velocity  
Suppose that the body moved from A to B in a time t.  
The linear speed*, v, of the body is given by v = s/t.  
If we divide the equation for angular displacement by t, we have  
 
We now define the quantity θ/t to be the angular velocity associated with this motion.  
 
The units of angular velocity are therefore rads-1 and the relation between v and ω is  
 
   
* we now sometimes include the word "linear" when talking about what we previously just called speed, to make sure there is no confusion between linear speed and angular speed.  
   
Angular Acceleration  
In the previous section we assumed that the body moved from A to B with constant speed.  
If the linear speed of the body changes then, obviously, the angular speed (velocity) also changes.  
We now define the angular acceleration, a, to be the rate of change of angular acceleration.  
So, if the angular velocity changes uniformly from ω1 to ω2 in time t, then we can write  
 
Recall that linear acceleration is given by  
 
Using relation between angular and linear velocity (above) we have  
 
So the relation between angular and linear acceleration is  
 
   
Time Period  
The time period of a circular motion, T is the time taken for one revolution.  
   
Rotational Frequency  
The rotational frequency, f of a circular motion is the number of revolutions (or orbits) per unit time.  
   
Thus the relation between time period and frequency is  
 
Also, since ω = θ/t, putting in the values corresponding to one revolution (one orbit) we have another useful relation  
 
or, in terms of frequency  
 
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