A-level Physics/Forces, Fields and Energy/Motion in a circle

Motion in a circle is a very interesting concept, and not very complicated either. There are many lines (in the non literal sense) that can be drawn between circular motion and linear motion. In fact, as you progress, you will find circular motion much more convinient that linear motion, because of some basic properties, most importantly, that the angular motion, of a body is the same for all particles, though their velocities may change. However, this will be dealt in rotational mechanics, not here. Before reading this section, ensure that you have a thorough understanding of linear motion, vectors and differentiation.

Angular Variables
Similar to the variables, $$ \vec r, \vec s , \vec v , \vec a $$ found in linear motion, representing the position vector, displacement, velocity and acceleration respectively, we have a few terms in angular motion.

The first variable is $$\ \theta $$, which is the angle subtended at the centre of the circle. This can be compared with the position vector $$ \vec r $$ of linear motion. It is measured in radians, or rads.

The second variable is angular velocity, $$\ \vec \omega $$. Like velocity is the change in your position vector, or your displacement by time, t, $$\ \vec \omega $$ is the change in angle per unit time. It is measured in radians per second, rads/s. Also, it is not your displaced angle. If you cover 360 &deg, and full circle, in one second, it does not mean that your angular velocity is zero, but 2 &pi radians per second. Mathematically we have, $$\ \vec \omega = d \theta / dt $$

The third variable is angular acceleration, $$\ \vec \alpha $$. It is the change in angular velocity by time. It is measured in radians per second, $$rads/s^2$$. Mathematically, $$\ \vec \alpha = d \omega / dt $$

Notice that these quantities are not dependent on radius. All angular terms depend only on the axis of rotation, or the centre of the circle, a fact that makes circular motion useful.

Axial Vectors
It should be noted that these vectors are not normal vectors, but are axial vectors. Axial vectors are vectors along the axis. Rather that along the direction of motion, these angular variables are along the axis, in an upward direction or downward direction.

This concept is quite difficult to visualise. Imagine a rod, which is your axis of rotation, passing through a disc. If you try and spin the disc, the axis will start to rotate. As such, your axis does not possess a real velocity. It does not move at all.

Now if you put a small ring on the rod. It should be in contact, but not too tightly attached. If you spin the rod, the ring will start to move up, or down. This is due to a physical phenomenon, but for this purpose, ignore the dynamics of its motion, only consider that it is moving up, or down. Also notice, that generally, when you rotate it in anticlockwise direction, it moves  up . By this experiment, you can visualise how the axial vector operates.

By convention, an anticlockwise rotation the direction of the axial vector is taken as the positive upward vector on the axis, and vice versa for clockwise rotation. Another point to note is that though axial vectors can be resolved, to simulate a body rotating in two axes, it more often than not complicates the situation. There are also several technical complications if your two axes of rotation are not passing through the same point. This is a very complicated situation, and will not be discussed.

Using Angular Variables
Here are a few examples showing the usage of the angular variables we have just learnt.

Example 1
Suppose a body is rotating, such that it subtends an angle of 1200 &deg at the centre every minute. Find its angular velocity in S.I. units.

We know that angular velocity is the angle covered per unit time. Since it covers 1200 degrees per minute, with uniform angular velocity, we can say that it covers 20 degrees in one second. 20 degrees is $$ \pi / 9 $$ radians. So we get $$\ \vec \omega = \pi / 9 $$ rad/s.

Example 2
If a body's angular displacement increases by $$ \pi $$ per second, find its, angular velocity, and acceleration at some time, t.

It is clear that the angular velocity is not constant from this. The average angular velocity in the first second is $$ \pi $$ rads/s, in the second second, $$ 2\pi $$ rads/s, and in the third second $$ 3\pi $$ rads/ sec. You can observe that the angular velocity is the time, t into $$ \pi $$ radians per second. So, $$ \vec \omega = t \pi $$ rads/s.

We can also see that the angular acceleration is constant, and equal to $$ \pi $$ radians per second square. So, $$ \vec \alpha = \pi $$.

Equations of Motion
We now move onto a few equations, bearing striking resemblance to those of linear motion. The second example given above, is much better solved with these equations. All these equations are applicable only under constant angular acceleration.

  $$ \vec \omega = \vec \omega_0 + \vec \alpha t $$ This equation gives a relation between your angular velocity and time. $$ \vec \omega_0 $$ is the angular velocity initially. 

$$ \theta = \theta_0 + \vec \omega_0 t + \frac{1}{2} \vec \alpha t^2 $$ This equation gives a relation between your angular displacement and time. $$ \theta_0 $$ is the initial angular displacement. 

 $$ \omega^2 - \omega_0^2 = 2 \theta \alpha $$ This equation gives a relation between your angular velocity and angular displacement. Remember that the &omega is not a vector. 



Example 3
If a body spins about an axis, accelerating at a rate of 4 rad/s^2, find
 * 1) the angular displacement after 5 seconds, and angular velocity at that time
 * 2) the angular displacement when it attains an angular velocity of 12 rad/s



 The time has been given. In the first part, we need a relation between &theta and time. This is the second equation. So, we have the equation identified, $$ \theta = \theta_0 + \vec \omega_0 t + \frac{1}{2} \vec \alpha t^2 $$. We also know the values of $$ \theta_0, \omega_0 and \alpha $$. Substituting, $$ \theta = 0 + 0*5 + \frac{1}{2} 4 5^2 = 2 \times 25 $$ $$ \theta = 50 $$ radians

The second part requires us to establish a relation between &omega and time. This is the first equation.

$$ \vec \omega = \vec \omega_0 + \vec \alpha t = 0 + 4 \times 5 = 20 $$ $$ \vec \omega = 20 $$ rads/s



 There are two methods to solve this equation. One is to find time through the first equation, and substitute it in the second, the other is to directly use the third equation. $$ \vec \omega = \vec \omega_0 + \vec \alpha t $$ $$ 12 = 0 + 4 \times t $$ or $$ t = 3 $$

Substituting in equation 2,

$$ \theta = 0 + 1/2 \times 4 \times 3^2 = 2 \times 9 = 18 $$

In the other method,

$$ 12^2 = 2\times \theta \times 4 $$ or $$ \theta = \frac {144}{2\times4} = 18 $$  

One might ask why the first method was even considered. This is because, if the angular velocity was given, not your speed, the third equation would require us to first find the speed, i.e. magnitude, of the &omega, and we would proceed further. This too is not a serious impediment, and could be carried out. But if the angular velocity were asked, the third equation would not give us that. These are important things to be kept in mind, even if they are not applied often.

Angular Unit Vectors
For the sake of convenience, two different vectors are used in circular motion, radial vectors and tangential vectors. Rather than our usual $$ \hat i $$ and $$ \hat j $$ vectors used for components in the x-axis and y-axis, the radial vector, $$ \hat r $$ gives the outward component along the radial line. The tangential vector, $$ \hat \tau $$ gives the component along the tangent, anticlockwise being positive.

If the angle subtended at the centre is known, then it becomes quite easy to convert these vectors into normal x-axis and y-axis vectors, by trigonometry.