Let us consider a point A in a two-dimensional Cartesian co-ordinate system, the position of this point can be described by a specific single vector. This vector is the displacement of that point A with respect to the origin O of the co-ordinate system, as shown in the figure below.

This vector is known as the “position vector” of that point A with respect to the origin O.

Here the position vector is denoted by the \( \vec{OA}=\vec{r} \).

\( \vec{r} \) gives the magnitude as well as the direction of displacement.

If \( \hat{r} \) be the unit vector along \( \vec{r} \), then we can write,

\( \displaystyle{\hat{r}=\frac{\vec{r}}{|\vec{r}|}}\tag{1} \). where, \( |\vec{r}| \) is the magnitude of the vector \( \vec{r} \).

So the expression for the velocity is

\( \vec{v}=\vec{v_r}+\vec{v_{\theta}} \)

\( Or,\ \vec{v}=\dot{r}\hat{r}+r\dot{\theta}\hat{\theta}\tag{2} \)

[ to know the derivation for the expression of velocity, CLICK HERE ]

Expression for acceleration:

Since the acceleration is the time derivative of the velocity \( \vec{v} \),

Therefore, acceleration \( \displaystyle{\vec{a}=\frac{d\vec{v}}{dt}} \).

or, \( \displaystyle{\vec{a}=\frac{d}{dt}(\dot{r}\hat{r}+r\dot{\theta}\hat{\theta})} \)

or, \( \vec{a}=\ddot{r}\hat{r}+\dot{r}\frac{d\hat{r}}{dt}+\dot{r}\dot{\theta}\hat{\theta}+r\ddot{\theta}\hat{\theta}+r\dot{\theta}\frac{d\hat{\theta}}{dt} \)

\(or,\ \vec{a}=\ddot{r}\hat{r}+\dot{r}\frac{d\hat{r}}{d\theta}\frac{d\theta}{dt}+\dot{r}\dot{\theta}\hat{\theta}+r\ddot{\theta}\hat{\theta}+r\dot{\theta}\frac{d\hat{\theta}}{d\theta}\frac{d\theta}{dt}\tag{3} \)

We know that, \( \hat{r}=\cos\theta\ \hat{i}+\sin\theta\ \hat{j} \)

and, \( \hat{\theta}=-\sin\theta\ \hat{i}+\cos\theta\ \hat{j} \)

[ To know the derivations of expression of \( \hat{r} \) and \( \hat{\theta} \) in terms of the unit vectors \( \hat{i} \) and \( \hat{j} \), CLICK HERE ].

Now derivative of \( \hat{r} \) with respect to \( \theta \) is

\( \displaystyle{\frac{d\hat{r}}{d\theta}=-\sin\theta\ \hat{i}+\cos\theta\ \hat{j}} \)

\(or,\ \displaystyle{\frac{d\hat{r}}{d\theta}=\hat{\theta}}\tag{4} \)

Again derivative of \( \hat{\theta} \) with respect to  \( \theta \) is

\( \frac{d\hat{\theta}}{d\theta}=-\cos\theta\ \hat{i}-\sin\theta\ \hat{j} \)

\(or,\ \frac{d\hat{\theta}}{d\theta}=-\hat{r}\tag{5} \).

Using the equation (4) & (5) in equation (3) we get,

\( \vec{a}=\ddot{r}\hat{r}+\dot{r}\dot{\theta}\hat{\theta}+\dot{r}\dot{\theta}\hat{\theta}+r\ddot{\theta}\hat{\theta}-r{\dot{\theta}}^2\hat{r} \)

\( or,\ \vec{a}=[\ddot{r}-r{\dot{\theta}}^2]\hat{r}+[r\ddot{\theta}+2\dot{r}\dot{\theta}]\hat{\theta}\tag{6} \)

Components of acceleration:

Equation (6) can be written as,

\( \vec{a}=\vec{a_r}+\vec{a_{\theta}} \)

where, the radial acceleration is \( \vec{a_r}=[\ddot{r}-r{\dot{\theta}}^2]\hat{r} \)

and the transverse acceleration \( \vec{a_{\theta}}=[r\ddot{\theta}+2\dot{r}\dot{\theta}]\hat{\theta} \)  

The magnitude of radial acceleration is \( |\vec{a_r}|=\ddot{r}-r{\dot{\theta}}^2 \)

and the magnitude of the trasverse acceleration is \( |\vec{a_{\theta}}|= r\ddot{\theta}+2\dot{r}\dot{\theta} \).

Here, the radial component of acceleration \( |\vec{a_r}| \) has two parts:

• First part is \( \ddot{r} \), which indicate the acceleration due to the change of in magnitude of \( \dot{r} \) with respect to time. It is directed away from the centre and has a positive sign.
• Second part is \( r{{\dot\theta}}^2 \), which indicates the centripetal acceleration due to the change in the magnitude of \( \theta \) with respect to time. It is directed towards the centre and has a negative sign.

Again the radial component of acceleration \( |\vec{a_{\theta}}| \) also have two parts:

• First part is \( r\ddot{\theta} \), which indicates the acceleraton due to the change in \( \dot{\theta} \) with respect to time.
• Second part is \( 2\dot{r}\dot{\theta} \) and it is similar to coriolis acceleration. This part comes into play as a result of interaction of linear and angular veocities due to the change in \( r \) and \( {\theta} \) with respect to time.