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What is the Definition of a Rigid Body? What is the Work Done by a Couple?
Rotation of Rigid Bodies
A rigid body is a continuous distribution of particles of defi-nite masses within an extended volume; these particles do not change their relative positions with respect to one another while rotating about a fixed axis.
Let us consider a rigid body R [Fig.]. The constituent particles of the body are A, B, C, …
For a pure rotational motion of this rigid body, each constituent particle undergoes a circular motion. The characteristic of these circular motions is that, the centres of all the circles lie on a single fixed straight line.
This line is normal to the plane of each of the circular paths and is called the axis of rotation of the rigid body.
Definition: The axis of rotation is defined as a fixed straight line that passes normally through the centre of the circular path followed by a rotating particle, or by any constituent particle of a rotating rigid body.
The axis of rotation may pass through the body itself, or may lie entirely outside the body.
Couple : Torque
Definition: Two equal, parallel and opposite forces, hav-ing different lines of action, acting simultaneously on a body, constitute a couple.
In Fig., a pair of parallel and opposite forces F, act on a body at points A and B. Then, (F, F) is a couple acting on
the body. The perpendicular distance between the lines of action of the two forces is called the arm of the couple. A couple tends to set up a rotational motion.
Definition: The tendency of rotational motion, set up in a body by a couple, is called the moment of the couple or torque. The torque is given by the product of one of the forces of the pair and the arm of the couple.
The direction of the torque is given by the direction of advance of a right-handed screw when turned in the direction of rotation.
Turning on or off a water-tap, screwing or unscrewing the cap of a bottle, using a screw driver, etc., are associated with torques applied with our fingers thereby setting up a rota-tional motion.
Vector form: Torque is a vector quantity. The vector representation for the relation between torque and force is
\(\vec{\tau}\) = \(\vec{r}\) × \(\vec{F}\)
Torque and the vector product
Torque is the vector product between the force vector \(\vec{F}\) and vector \(\vec{r}\)
\(\vec{\tau}\) = \(\vec{r}\) × \(\vec{F}\)
Unit and dimension of torque
= dimension of force × dimension of length = MLT-2 × L = ML2T-2
Torque and pure rotation: Torque due to a couple can produce rotational motion only. In Fig, the resultant of the two forces applied at points A and B is zero, i.e., F – F = 0 . As the resultant force is zero, so no change occurs in its translational motion. But their lines of action are separate. So, only rotation is set up in this case. Such a rotation without translation is called pure rotation.
Moments of the two forces of a couple: A point O is taken on the line BC as shown in Fig. Moment of force F applied at point A with respect to O = F × CO. Also, force F at point B sets up the moment about O as F × BO. Flence, algebraic sum of these two moments = F × CO + F × BO = F(CO + BO) = F × BC. But F × BC is the torque generated by the couple.
The algebraic sum of moments of the two forces of a cou-ple, about a point, is equal to the moment of the couple (often called torque) about that point.
Moment of force and torque are identical: From the above discussion, it is clear that a torque alone can pro-duce pure rotational motion. But it is often observed that, pure rotation can also be produced by a single force only. For example, a door can be opened by applying a single force on the door panel. However, even in this case, an equal and opposite reaction force on the door panel is developed at the hinges. This reaction, along with the applied force, constitutes a couple, and exerts a torque.
Hence, moment of a force and torque are two identical physical quantities.
Work done by a Couple
We know that, two forces of equal magnitude constitute a couple. When a couple produces rotation in a body, the sum of the work done by the two forces of the couple is the
measure of the work done by the couple. Suppose a couple (F, F) is acting on a body [Fig.],
AB is the arm of this couple. So, the moment of the couple or torque, r = F × AB.
Suppose the body is rotated by an angle θ under the influ-ence of the couple about the point O. As a result, the point A is shifted to A1 and B to B1. If θ is very small, then we can assume that the arcs AA1 and BB1 are almost straight lines.
Now, AA1 = AO ᐧ θ and BB1 = BO ᐧ θ
Work done by the force acting on the point A,
W1 = F ᐧ AA1 = F ᐧ AO ᐧ θ
Similarly, work done by the force acting on the point B,
W2 = F ᐧ BB1 = F ᐧ BO ᐧ θ
So, total work done by the couple,
W = W1 + W2 = F ᐧ AO ᐧ θ + F ᐧ BO ᐧ θ
= F ᐧ (AO + BO) ᐧ θ = F ᐧ AB ᐧ θ = r-8
= torque × angular displacement
Hence, the amount of work done does not depend on the position of axis of rotation. For one complete rotation of the body, the angular displacement is 2π. So, for n complete rotations the angular displacement will be 2 π n.
Hence, for n complete rotations the work done by the couple,
W = 2πn × torque