Lecture 11
Motion in a plane: Introduction to polar coordinates

So far we have discussed equilibrium of bodies i.e. we have concentrated only on statics. From this lecture onwards we learn about the motion of particles and composite bodies and how it is affected by the forces applied on the system. Thus we are now starting study of dynamics.

When we describe the motion of a particle, we specify it by giving its position and velocity as a function of time. How the motion changes with time is given by the application of Newton's II^nd Law. One such particle at position moving with velocity and acted upon by a force is shown in figure 1. The force gives rise to an acceleration . Notice that in general the position, the velocity and the acceleration are not in the same direction.

Each of these vectors is specified by giving its component along a set of conveniently chosen axes. For a particle moving in a plane, if we choose the Cartesian coordinate system (x-y axes) then the position is given by specifying the coordinates (x, y), velocity by its components and acceleration by its components . These are related by the relationship

and

These expressions are easily generalized to three dimensions by including the z-component of the motion also. However, in this lecture we will be focusing on motion in a plane only. With these components the equations of motion to be solved are

Coupled with the initial conditions solutions of these equations provide the velocity and position of a particle uniquely. However, the Cartesian system of coordinates is only one way of describing the motion of a particle. There arise many situations where describing the motion in some other coordinate system i.e., taking components along some other directions is move convenient. One such coordinate system is polar coordinates. In this lecture we discuss the use of this system to describe the motion of a particle. To introduce you to polar coordinates and how their use may make things easy, we start with the discussion of a particle in a circle.

Consider a particle is moving with a constant angular speed ω in a circle of radius R centered at the origin (see figure 2). Its x and y coordinates are given as

with both x and y being functions of time (see figure 2).

On the other hand, if we choose to give the position of the particle by giving its distance r from the origin and the angle Φ that the line from the origin to the particle makes with x-axis in the counter-clockwise direction, then the position is given as

In this coordinate system, r is a constant and Φ a linear function of time. Thus there is only one variable that varies with time whereas the other one remains constant. The motion description thus is simpler. These co-ordinates are known as the planar polar coordinates. As expected, these coordinates are most useful in describing motion when there is some sort of a rotational motion. We will therefore find them useful, for example, in discussing motion of planets around the sun rotating bodies and motion of rotating objects.

So to start with let us set up the unit vectors is polar co-ordinates ( r, Φ ) . Given a point , the unit vector is in outward radial direction and has magnitude of unity. The Φ unit vector is also of magnitude unity and is perpendicular to and in the direction of increasing Φ (see figure 3). Obviously the dot product . In term of the unit vectors in x and y direction these are given as

As is clear from these expression the direction of and Φ is not fixed but depends on the angle Φ. On the other hand, it does not depend on r. If we go along a radius, and Φ remain unchanged as we move (recall that two parallel vectors of same magnitude are equal). But that is not the case if Φ is changed.

The position a of a particle in polar co-ordinates to given by writing

As particle moves about, changes. Does the mean that the velocity

The answer is no. As already discussed is a function of Φ, the angle from the x-axis. Thus as a particle moves such that the angle Φ changes with time, the unit vector also changes. Its derivative with respect to time is therefore not zero. Thus the correct expression for is

Let us now calculate . As already stated, does not change as one moves radically in or out. Thus changes only if Φ changes. Let us now calculate this change (figure 4)

As is clear from the figure

where the dot on top of a quantity denotes its time derivative. The expression above can also be derived mathematically as follows:

Thus the velocity of a particle is given as

We note that the unit vectors in polar coordinates keep changing as the particle moves because they are given by the particles current position. Thus even if a particle were moving with a constant velocity, the components of velocity along the radial and the directions will change. Let us calculate the velocity of a particle moving in a circle with a constant angular speed. For such a particle

so the velocity is given as

This is a well known result: the velocity of a particle moving in a circle with a constant angular speed is in the tangential direction and its magnitude is Rω. How about the acceleration in polar coordinates? This is the derivative of with respect to time. Thus

As was the case with the unit vector , the unit vector also is a function of the polar angle Φ and as such changes as the particle moves about. Thus in calculating the acceleration, time derivative of also should be taken into account. From figure 4 it is clear that

This can also be derived mathematically as

Using this derivative and the chain rule for differentiation, we get

You can see that the expression is a little complicated. The complexity of the expression arises because the unit vectors are changing as the particle moves. You can check for yourself that for a particle moving with a constant velocity, the expression above will give zero acceleration. Despite little complicated expressions for the acceleration, employing polar coordinates becomes really useful in situations where motion is circular-like as we will see in two standard examples later. Let us first go to one familiar example of a particle moving in a circle for which r = R , . This gives

which is the correct answer for the centripetal acceleration. For this reason is known as the centripetal term. Let us now solve an example of mechanics using polar co-ordinates.