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
ω_{1} to ω_{2}
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 


