18.3 Rotation  (Page 5/5)

$$\begin{array}{l}\Rightarrow {\tau }_x={\tau }_1+{\tau }_2=2-3=-1Nm\end{array}$$

The net rotational torque is negative and hence clockwise in direction. The magnitude of net torque about x-axis is :

$$\begin{array}{l}\Rightarrow {\tau }_x==1Nm\end{array}$$

Now, we proceed to calculate torque about the origin of reference which lies on the axis of rotation. The torque about the origin of coordinate system is given by :

$$\begin{array}{l}\mathbf{\tau }=\mathbf{r}\mathbf{x}\mathbf{F}\end{array}$$

τ = | i j k | | 1 1 -1 || 2 2 -3 |

$$\begin{array}{l}\Rightarrow \mathbf{\tau }=\left[\right(1x-3)-(2x-1\left)\right]\mathbf{i}+\left[\right(-1x2)-(1x-3\left)\right]\mathbf{j}+\left[\right(1x2\}-(1x2\left)\right]\mathbf{k}\\ \Rightarrow \mathbf{\tau }=(-3+2)\mathbf{i}+(-2+3)\mathbf{j}\\ \Rightarrow \mathbf{\tau }=-\mathbf{i}+\mathbf{j}\end{array}$$

The particle, here, moves about the axis of rotation in x-direction. In this case, the particle is restrained not to rotate about any other axis. Thus, torque in rotation about x-axis is equal to the x- component of torque about the origin. Now, the vector x-component of torque is :

$$\begin{array}{l}{\mathbf{\tau }}_x=-\mathbf{i}\end{array}$$

The net rotational torque is negative and hence clockwise in direction. The magnitude of torque about x-axis for rotation is :

$$\begin{array}{l}{\tau }_x=1Nm\end{array}$$

Nature of motion

Nature displays varieties of motion. Most of the time, we encounter motion, which is composed of different basic forms of motion. The most important challenge in the study of motion is to establish a clear understanding of the components (types) of motion that ultimately manifest in the world around us. Translational motion is the basic form. It represents the basic or inherent property of natural objects. A particle moving in straight line keeps moving unless acted upon by external force. However, what we see around is not what is fundamental to the matter (straight line motion), but something which is grossly modified by the presence of force. This is the reason planets move around the Sun; electrons orbit about nucleus and aero-planes circle the Earth on inter-continental flight.

Variety of motion is one important aspect of the study of motion. Another important aspect is the restrictive paradigm of fundamental laws that needs to be expanded to real time bodies and motions. For example, Newton's laws of motion as postulated are restricted to particle or particle like bodies. It is required to be adapted to system of particles and rigid body systems with the help of concepts like center of mass. In real time, motions may be composed of even higher degree of complexities. We can think of motions which involve rotation while translating. Now, the big question is whether Newtonian dynamics is capable to describe such composite motions? The answer is yes. But, it needs further development that addresses issues like rotational dynamics of particles and rigid bodies. We shall consider this aspect of rotational motion in the next module titled Rotation of rigid body .

Summary

1. The particles in a rigid body are locked and are placed at fixed linear distances from others.

2. Each particle constituting a rigid body executes circular motion about a fixed axis in pure rotation.

3. The cause of change in angular velocity in rotation is torque. It is defined as :

$$\begin{array}{l}\mathbf{\tau }=\mathbf{r}\mathbf{x}\mathbf{F}\end{array}$$

where position vector," r ", and the force vector," ( F )" are in the plane of rotation of the point of application of force, which is perpendicular to the axis of rotation.

4. The magnitude of torque

The magnitude of torque is determined in three equivalent ways :

(i) In terms of angle between position vector ( r ) and force ( F )

$$\begin{array}{l}\tau =rF\sin \theta \end{array}$$

(ii) In terms of tangential component of force ( $F_T$ or $F_{\perp }$ )

$$\begin{array}{l}\tau =r\left(F\sin \theta \right)=r{F}_{\perp }\end{array}$$

(iii) In terms of moment arm ( $r_{\perp }$ )

$$\begin{array}{l}\tau =\left(r\sin \theta \right)F=r_{\perp }F\end{array}$$

4. Direction of torque

The vector equation of torque reveals that torque is perpendicular to the plane formed by position and force vectors and is also perpendicular to each of them individually. In order to know the sign of torque, we apply right hand rule (positive for anticlockwise and negative for clockwise rotation).