Rotational equivalent of force

We know that a rigid body, can have two kinds of motion, linear and rotational. Though, in general, the motion of an object is a combination of the two. But if we fix the body at a point then it is left with rotational motion only. We all know form Newton’s 2nd law that, to produce linear acceleration, we need something called force. Here one question may arise, that what is that thing which produces rotational acceleration? Before answering that let us take a simple example of opening of a door. The door is fixed with its hinges which can rotate about the axis of the hinges. But it should be noted that mere force cannot produce any rotation here. If we apply force on a point near the hinge, it is relatively difficult to rotate the door than when the same magnitude of force is applied on a point near to the end of the door. From the above experience it is clear that distance from the axis of rotation is also important here in order to produce any rotation. And not only that, the applied force must be in a direction which is perpendicular to the motion of the door. So, it is not only force but where the force is applied and which direction it is applied is also, important here. Now to answer to the question, the thing which produces rotational acceleration is called as moment of force. Sometimes, it is also called torque and sometimes couple.

When, forceis acting on a body at P of which  is the position vector w.r.t the origin O. then, the mathematical form of moment of force is given by

……………………………..(1)

The direction of   will be perpendicular to the plane containing both and  . The magnitude of torque is given by

Here   is the angle between  and  .

The dimension of force is   and that of torque is 

From equation (1), it is clear that the value of   is zero if either  or  equals to zero or the line of action of the force is along .  has maximum value when line of action is perpendicular to Thus, answering to our question of opening of the door.

Here the point to remember is that if the direction of  either or    is reversed, then the direction of  is also reversed. But if the direction of both of them is reversed, then the direction of  remains the same.

Similar to torque, the rotational equivalent of linear momentum is called angular momentum, which is defined as

.

Hence,

Q> The torque on a body about a given point is given by  . Here is a constant vector and  is the angular momentum. Then,

  1. is perpendicular to  .
  2. is a constant of time
  3. The component of along  does not changes with time.
  4. The magnitude of does not change with time but its direction does.

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January 23, 2018

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