A rigid body in rotation about a fixed axis should continue to rotate with the given angular velocity indefinitely unless obstructed externally. The angular velocity of rotation is changed by external cause in the same manner as in translational motion. In the case of translation motion, the external cause is "force". We have to investigate what is the equivalent "cause" in the case of rotational motion?

Let us consider a rigid block placed over a smooth horizontal surface as shown in the figure, which is subjected to a force across one of its face. What is expected? The center of mass will move with linear acceleration following Newton's second law. However, the force is not passing through the center of mass. As such, the face of the block will also have turning tendency in the direction shown.

The motion of block |
---|

Let us now imagine if the body is pierced through in the middle by a vertical bar with some clearance. The vertical bar will inhibit translation and the block will only rotate about the vertical bar. The question is what caused the block to turn around? Indeed it is the force that caused the angular motion. However, it is not only the force that determined the outcome (magnitude and direction of angular velocity and acceleration). There are other considerations as well.

Rotational motion |
---|

We all have the experience of opening and closing a hinged door in our house. It takes lesser effort (force) to open the door, when force is applied farther from the hinge. We can further reduce the effort by applying force normal to the plane of the door. On the other hand, we would require greater effort (force), if we push or pull the door from a point closer to the hinge. If we apply force in the radial direction (in the plane of door) towards the hinge, then the door does not rotate a bit - whatever be the magnitude of force. In the nutshell, rotation of the door depends on :

- Magnitude of force
- Point of application of force with respect to hinge (axis of rotation)
- Angle between force and perpendicular line from the axis of rotation

These factors, which cause rotation, are captured by a quantity known as torque, which is defined as :

where "r" is the position vector. There is a slight difficulty in interpreting this vector equation as applicable to rotation. The position vector,"r", as we can recall, is measured from a point. Now, the question is what is that point in the case of rotation about a fixed axis? Here, we observe that there is a unique point of application of the force. By the nature of motion, this point rotates in a plane perpendicular to the axis of rotation. We can, therefore, uniquely define the position of the point of application of force by measuring it from the point on the axis in that plane of rotation.

Rotational motion |
---|

The force may be directed in any possible direction. However, the body rotates about a fixed axis. It is not allowed to rotate or move about any other axes. What it means that we are limited only to the component of torque along the axis of rotation. It, then, means that we are only interested in components of force that lie in the plane of rotation, which is perpendicular to the axis of rotation. This we achieve by ensuring to only consider the components of force which are perpendicular to the axis of rotation i.e. in the plane of rotation. In the figure below, we have tried to capture this aspect. We consider only the component of force in xy-plane while evaluating the vector torque equation given above.

Rotational motion |
---|

In the nutshell, we interpret the vector product with two specific conditions :

- The position vector (r) is measured from a point on the axis, which is in the plane of rotation.
- The force vector (F) is component of force in the plane of rotation.

The rotation of the rigid body about an axis passing through the body itself is composition of very large numbers of circular motions by particles composing it. Each of the particles undergo circular motion about the axis in a plane which is perpendicular to it.

** Magnitude of torque **

With the understandings as deliberated above, let us determine the magnitude of torque. Here, let us consider that a force is applied at a point which is at a perpendicular distance "r" from the axis and in the plane of rotation. As far as application of force is concerned, it is a force applied at a point in the plane of rotation. In order to keep the context simplified, we have not drawn the rigid body assuming that the point of application rotates in a circle whose plane is perpendicular to the axis of rotation. Now, the magnitude of torque is given as :

Torque |
---|

where "r","F" and "θ" are quantities measured in the plane perpendicular to the axis of rotation.

Equivalently, we can determine the magnitude of torque as :

The magnitude of torque is equal to the product of the magnitude of position vector as drawn from the center of the circle and component of force perpendicular to the position vector. This is an important interpretation as it highlights that it is the tangential or perpendicular component (perpendicular to radial direction) of force, which is capable to produce change in the rotation. The component of force in the radial direction (F cosθ) does not affect rotation. We can also rearrange the expression of magnitude as :

Also,

The magnitude is equal to the product of the perpendicular distance and magnitude of the force. Perpendicular distance is obtained by drawing a perpendicular on the extended line of application of force as shown in the figure below. We may note here that this line is perpendicular to both the axis of rotation and force. This perpendicular distance is also known as "moment arm" of the force and is denoted as "

Torque |
---|

** Direction of torque **

The nature of cross (or vector) product of two vectors, conveys a great deal about the direction of cross product i.e. torque where both position and force vectors are in the plane of rotation. It tells us that (i) torque vector is perpendicular to the plane formed by operand vectors i.e. "r" and "F" and (ii) torque vector is individually perpendicular to each of the operand vectors. Applying this explanation to the case in hand, we realize here that torque is perpendicular to the plane formed by radius and force vectors i.e z-axis. However, we do not know which side of the plane i.e +z or -z direction, the torque is directed.

Direction of torque |
---|

We apply right hand rule to determine the remaining piece of information, regarding direction of torque. We have two options here. Either we can shift radius vector such that tails of two vectors meet at the position of particle or we can shift the force vector (parallel shifting) so that tails of two vectors meet at the axis. Second approach has the advantage that direction of torque vector along the axis also gives the sense of rotation about that axis. Thus, following the second approach, we shift the force vector to the origin, while keeping the magnitude and direction same as shown here.

Direction of torque |
---|

Now, the direction of rotation is obtained by applying rule of vector cross product. We place right hand with closed fingers such that the curl of fingers point in the direction as we transverse from the direction of position vector (first vector) to the force vector (second vector). Then, the direction of extended thumb points in the direction of torque. Alternatively, we see that a counter clock-wise torque is positive, whereas clock-wise torque is negative. In this case, torque is counter-clockwise and is positive. Therefore, we conclude that torque is acting in +z-direction.

Pure rotation about a fixed axis gives us an incredible advantage in determining torque. We work with only two directions (positive and negative). In the case of torque about a point, however, we consider other directions of torque as well. It is also noteworthy to see that torque follows the superposition principle. Mathematically, we can add torque vectors algebraically as there are only two possible directions to obtain net or resultant torque. In words, it means that if a rigid body is subjected to more than one torque, then we can represent the torques by a single torque, which has the same effect on rotation.

Torque has the unit of Newton - meter.