20.2 Second law of motion in angular form  (Page 2/2)

$$\begin{array}{l}\Rightarrow {\mathbf{F}}_{\mathrm{net}}=m\mathbf{a}\end{array}$$

Substituting the same in the equation above, we have :

$$\begin{array}{l}\Rightarrow \frac{d\mathbf{\ell }}{dt}=\mathbf{r}\mathbf{x}m\mathbf{a}=\mathbf{r}\mathbf{x}{\mathbf{F}}_{\mathrm{net}}=\mathbf{r}\mathbf{x}\sum {\mathbf{F}}_i\end{array}$$

We note here that all forces are acting at the same point (occupied by the particle). As such, position vectors for all forces are same. Thus, we can enclose the vector product inside the summation sign,

$$\begin{array}{l}\Rightarrow \frac{d\mathbf{\ell }}{dt}=\sum \mathbf{r}\mathbf{x}{\mathbf{F}}_i=\sum {\mathbf{t}}_i={\mathbf{t}}_{\mathrm{net}}\end{array}$$

In words, Newton's second law of motion for a particle in angular form is read as :

Newton's second law of motion in angular form

The net torque on a particle is equal to the time rate of change of its angular momentum.

The quantities torque and angular momentum are both measured or calculated about a common point "O". If it is not so, then the relationship is not valid. This aspect should always be kept in mind while applying the law.

We have emphasized that the angular form of Newton's second law is defined about a point - not an axis and as such not limited to rotational motion. Hence, we can employ this law even in situation where motion is strictly along a straight line. We shall, as a matter of fact, work out an example to illustrate this.

Newton's second law for a particle in rotation

Newton's law for a particle in rotation is established by taking first derivative of the angular momentum of a particle about an axis of rotation. As derived in previous module titled Angular momentum . it is given by :

Differentiating with respect to time,

$$\begin{array}{l}{t}_{\mathrm{net}}=\frac{d\ell }{dt}=\frac{d\left(\mathrm{I\omega }\right)}{dt}\end{array}$$

Since particle moves around a circular path of a constant radius, $I=m{r_{\perp }}^2$ is a constant and the same can be taken out of differentiation,

This derivation underlines that the above relation is specific to rotation about an axis - not for general motion. On the other hand, the definition of torque as the time rate of angular momentum about a point is a general form of Newton's second law.

Summary

Since discussion for angular momentum and corresponding Newton's second law are considered for different situations, it would be of great help to summarize the main results and special points for each of the cases so far covered. For clarity, we summarize as follows :

General motion of a particle

1: The reference for measurement of various angular quantities is a fixed point in the coordinate system. All angular quantities are measured about the same point.

2: Angular momentum of a particle in motion is given as :

$$\begin{array}{l}\mathbf{\ell }=\mathbf{r}\mathbf{x}\mathbf{p}=m(\mathbf{r}\mathbf{x}\mathbf{v})\end{array}$$

3: Newton's second law of motion in angular form for a particle is expressed mathematically as :

$$\begin{array}{l}{\mathbf{t}}_{\mathrm{net}}=\frac{d\mathbf{\ell }}{dt}\end{array}$$

In particular we must note that the form of Newton's second expressed in terms of moment of inertia and angular acceleration is not valid for general motion. It is so because, moment of inertia is defined about an axis - not about a point. For this reason, we shall not use the following form for general motion,

$$\begin{array}{l}{t}_{\mathrm{net}}=I\alpha \end{array}$$

Rotational motion of a particle

1: For rotational motion, the reference for measurement of various angular quantities is a fixed axis of rotation. All angular quantities are measured about the same axis of rotation.

2: Angular momentum of a particle in motion is given as :

$$\begin{array}{l}\ell =r_{\perp }p=m{r}_{\perp }v=m{r}_{\perp }x\omega r_{\perp }=m{r_{\perp }}^2\omega =I\omega \end{array}$$

where "I" and "ω" denote moment of inertia and angular velocity respectively. Here, " $r_{\perp }$ " is moment arm about the axis of rotation.

Since there are only two directions invoved (one in the direction of axis and other opposite to it), it is possible to represent the directional features of angular quantities in rotation with a preceding "plus" or "minus" sign for anti-clockwise and clockwise rotational motions respectively. This is also the reason why do we write equations of rotation in scalar form.

3: Newton's second law of motion is expressed mathematically as :

$$\begin{array}{l}{t}_{\mathrm{net}}=\frac{d\ell }{dt}=I\alpha \end{array}$$