7 CIRCULAR MOTION 7.5 Non-uniform circular motion.i e. =q
sin θ + cos^2 θ = 1. (7.35)r.e i θ. =q
cos^2 θ + sin^2 θ = 1. (7.33)Im(z)zy(^) x (^) Re(z)
Figure 62: Representation of a complex number in the complex plane.
Now, as we have just discussed, we can think of e i θ as representing a vector in
the complex plane: the real and imaginary parts of e i θ form the coordinates of
the head of the vector, whereas the tail of the vector corresponds to the origin.
What are the properties of this vector? Well, the length of the vector is given by
(^)
In other words, e i θ represen
.
ts a
.
unit vector. In fact, it is clear from Fig. 63 that e i θ
represents the radial unit vector er for an object whose angular polar coordinate
(measured anti-clockwise from the real axis) is θ. Can we also find a complex
representation of the corresponding tangential unit vector eθ? Actually, we can.
The complex number i e i θ can be written
i e i θ = − sin θ + i cos θ. (7.34)
Here, we have just multiplied Eq. (7.32) by i, making use of the fact that i^2 = − 1.
This number again represents a unit vector, since
i θ 2
Moreover, as is clear from
.
Fig.
.
63 , this vector is normal to e , in the direction of
increasing θ. In other words, i e i θ represents the tangential unit vector eθ.
Consider an object executing non-uniform circular motion in the complex
plane. By analogy with Eq. (7.26), we can represent the instantaneous position