A Classical Approach of Newtonian Mechanics

7 CIRCULAR MOTION 7.5 Non-uniform circular motion

er

Figure 61: Polar coordinates.

and θ. Here, r is the radial distance of the object from the origin, whereas θ is
the angular bearing of the object from the origin, measured with respect to some

arbitrarily chosen direction. We imagine that both r and θ are changing in time.

As an example of non-uniform circular motion, consider the motion of the Earth

around the Sun. Suppose that the origin of our coordinate system corresponds to

the position of the Sun. As the Earth rotates, its angular bearing θ, relative to the

Sun, obviously changes in time. However, since the Earth’s orbit is slightly ellipti-

cal, its radial distance r from the Sun also varies in time. Moreover, as the Earth

moves closer to the Sun, its rate of rotation speeds up, and vice versa. Hence, the

rate of change of θ with time is non-uniform.

Let us define two unit vectors, er and eθ. Incidentally, a unit vector simply a
vector whose length is unity. As shown in Fig. 61 , the radial unit vector er always

points from the origin to the instantaneous position of the object. Moreover, the

tangential unit vector eθ is always normal to er, in the direction of increasing θ.
The position vector r of the object can be written

r = r er. (7.26)

In other words, vector r points in the same direction as the radial unit vector er,

and is of length r. We can write the object’s velocity in the form

v = ̇r = vr er + vθ eθ, (7.27)

e

r

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