acts towards the centre of the circle. In other words,
T sin θ = m ω2r. (7.19)
Taking the ratio of Eqs. (7.18) and (7.19), we obtain tan θ = ω
2r
g . (7.20)
However, by simple trigonometry,
tan θ = r h. (7.21) Hence, we find ω = s g h. (7.22)
Note that if l is the length of the string then h = l cos θ. It follows that ω =
s g
l cos θ. (7.23)
For instance, if the length of the string is l = 0.2 m and the conical angle is θ = 30◦ then the angular velocity of rotation is given by
ω = v u u t 9.81 0.2 × cos 30◦ = 7.526rad./s. (7.24) This translates to a rotation frequency in cycles per second of
f = ω
2 π = 1.20Hz. (7.25)
7.5 Non-uniform circular motion
Consider an object which executes non-uniform circular motion, as shown in Fig. 61. Suppose that the motion is confined to a 2-dimensional plane. We can
7 CIRCULAR MOTION 7.5 Non-uniform circular motion
r
θ
er eθ
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
7 CIRCULAR MOTION 7.5 Non-uniform circular motion
whereas the acceleration is written
a = ˙v = arer + aθeθ. (7.28) Here, vr is termed the object’s radial velocity, whilst vθ is termed the tangential ve- locity. Likewise, ar is the radial acceleration, and aθ is the tangential acceleration. But, how do we express these quantities in terms of the object’s polar coordinates r and θ? It turns out that this is a far from straightforward task. For instance, if we simply differentiate Eq. (7.26) with respect to time, we obtain
v = ˙r er + r ˙er, (7.29) where ˙er is the time derivative of the radial unit vector—this quantity is non- zero because er changes direction as the object moves. Unfortunately, it is not entirely clear how to evaluate ˙er. In the following, we outline a famous trick for calculating vr, vθ, etc. without ever having to evaluate the time derivatives of the unit vectors er and eθ.
Consider a general complex number,
z = x + i y, (7.30)
where x and y are real, and i is the square root of −1 (i.e., i2 = −1). Here, x is the real part of z, whereas y is the imaginary part. We can visualize z as a point in the so-called complex plane: i.e., a 2-dimensional plane in which the real parts of complex numbers are plotted along one Cartesian axis, whereas the correspond- ing imaginary parts are plotted along the other axis. Thus, the coordinates of z in the complex plane are simply (x, y). See Fig. 62. In other words, we can use a complex number to represent a position vector in a 2-dimensional plane. Note that the length of the vector is equal to the modulus of the corresponding complex number. Incidentally, the modulus of z = x + i y is defined
|z| = qx2 + y2. (7.31)
Consider the complex number ei θ, where θ is real. A famous result in complex analysis—known as de Moivre’s theorem—allows us to split this number into its real and imaginary components:
7 CIRCULAR MOTION 7.5 Non-uniform circular motion Im(z) Re(z) x y z
Figure 62: Representation of a complex number in the complex plane.
Now, as we have just discussed, we can think of ei θ as representing a vector in the complex plane: the real and imaginary parts of ei 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
e i θ = q cos2θ +sin2θ = 1. (7.33)
In other words, ei θrepresents a unit vector. In fact, it is clear from Fig. 63that ei 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 ei θ can be written
iei θ = −sin θ + i cos θ. (7.34)
Here, we have just multiplied Eq. (7.32) by i, making use of the fact that i2 = −1. This number again represents a unit vector, since
ie i θ = q sin2θ +cos2θ = 1. (7.35)
Moreover, as is clear from Fig. 63, this vector is normal to er, in the direction of increasing θ. In other words, i ei 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
7 CIRCULAR MOTION 7.5 Non-uniform circular motion θ cos θ cos sin θ i θ i θ Im(z) Re(z) θ θ - sin eθ er i e e
Figure 63: Representation of the unit vectorserandeθin the complex plane.
vector of this object via the complex number
z = rei θ. (7.36)
Here, r(t) is the object’s radial distance from the origin, whereas θ(t) is its angu- lar bearing relative to the real axis. Note that, in the above formula, we are using ei θ to represent the radial unit vector e
r. Now, if z represents the position vector of the object, then ˙z = dz/dt must represent the object’s velocity vector. Differ- entiating Eq. (7.36) with respect to time, using the standard rules of calculus, we obtain
˙
z = ˙r ei θ+ r ˙θ iei θ. (7.37) Comparing with Eq. (7.27), recalling that ei θ represents er and i ei θ represents
eθ, we obtain
vr = ˙r, (7.38)
vθ = r ˙θ = r ω, (7.39)
where ω = dθ/dt is the object’s instantaneous angular velocity. Thus, as desired, we have obtained expressions for the radial and tangential velocities of the object in terms of its polar coordinates, r and θ. We can go further. Let us differentiate ˙
z with respect to time, in order to obtain a complex number representing the object’s vector acceleration. Again, using the standard rules of calculus, we obtain
7 CIRCULAR MOTION 7.5 Non-uniform circular motion
Comparing with Eq. (7.28), recalling that ei θ represents er and i ei θ represents
eθ, we obtain
ar = ¨r − r ˙θ2 =¨r − r ω2, (7.41) aθ = r ¨θ + 2˙r ˙θ = rω + 2˙ ˙r ω. (7.42) Thus, we now have expressions for the object’s radial and tangential accelerations in terms of r and θ. The beauty of this derivation is that the complex analysis has automatically taken care of the fact that the unit vectors er and eθ change direction as the object moves.
Let us now consider the commonly occurring special case in which an object executes a circular orbit at fixed radius, but varying angular velocity. Since the radius is fixed, it follows that ˙r = ¨r = 0. According to Eqs. (7.38) and (7.39), the radial velocity of the object is zero, and the tangential velocity takes the form
vθ = r ω. (7.43)
Note that the above equation is exactly the same as Eq. (7.6)—the only difference is that we have now proved that this relation holds for non-uniform, as well as uniform, circular motion. According to Eq. (7.41), the radial acceleration is given by
ar = −r ω2. (7.44)
The minus sign indicates that this acceleration is directed towards the centre of the circle. Of course, the above equation is equivalent to Eq. (7.15)—the only difference is that we have now proved that this relation holds for non-uniform, as well as uniform, circular motion. Finally, according to Eq. (7.42), the tangential acceleration takes the form
aθ = rω.˙ (7.45)
The existence of a non-zero tangential acceleration (in the former case) is the one difference between non-uniform and uniform circular motion (at constant radius).
7 CIRCULAR MOTION 7.6 The vertical pendulum