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Advanced Physics Topics like quantum mechanics and relativity have revolutionized our understanding of the universe.
Rigid Bodies: Translational Motion And Rotational Motion
Circular or Rotational Motion of a Particle
A material particle may possess two kinds of motion:
1. translational motion and
2. circular or rotational motion. Sometimes the particle may possess both kinds of motion simultaneously. This is known as a mixed motion. We have already discussed translational motion elaborately in the chapter on One-dimensional Motion. The circular motion of a particle will be discussed in this chapter.
Definition: If a particle is moving along a circular path about a point as the centre then the motion of that particle is called circular or rotational motion.
The surface in which the circular path lies is called the plane of rotation. A straight line drawn through the centre of the circular path and perpendicular to the plane of rotation is called the axis of rotation.
Let us consider a particle is revolving along a circular path of radius r and centre O [Fig.].
The perpendicular drawn through the point O on the plane of rotation is the axis of rotation [Fig.]. The radius r of the circle is called the radius vector whose direction is towards the particle away from the centre. During the revolution of the particle, the radius vector also keeps on rotating.
Angular coordinate: In Fig., let OP be the radius of the circle, and be taken as the reference for rotation i.e., at time t = 0, the position of the particle is at point P. Let us assume that the position of the particle is at the point A on the circumference of the circle after time t and the arc AP makes an angle θ1 with OP at the centre O. This angle θ1 is called the angular coordinate for the position A of the particle with respect to the radius OP. Similarly, the angular coordinates of the points B and C are θ2 and -θ3, respectively. So, the position of the particle on a particular circular path can be determined from its angular coordinate only. It should be noted that the angular coordinate for the position C in the given figure is negative. Usually, for anticlockwise rotational motion, θ is taken as positive and for clockwise motion, negative.
Angular Displacement
Definition: The angle subtended at the centre by the initial and final positions of a particle revolving in a circular path is called the angular displacement of the particle.
Let us consider that the initial position of a particle is the point A whose angular coordinate is θ1; and its final position is B whose angular coordinate is θ2 [Fig.]. So, for the movement of the particle from A to B, i.e., for its path of motion AB, the angular displacement is
θ = ∠AOB = θ2 – θ1 – change in angular position
So, the value of ∠AOB indicates the value of the angular dis-placement.
Measurement of Angles
The most commonly used unit for the measurement of angles is degree. Moreover, an angle can also be measured by the ratio of the arc-length subtending that angle at the centre of the circle to the radius of the circle.
For example, in the Fig., θ = ∠AOB \(=\frac{\text { arc length } A B}{\text { radius } O A}\)
Actually, the relation, θ = \(\frac{\text { arc }}{\text { radius }}\), provides the definition of an angle θ. Arc and radius—both have the dimension of length. So, their ratio, an angle is a dimensionless quantity. The unit of the angle, defined as \(\frac{\text { arc }}{\text { radius }}\), is radian (abbreviated as rad). Hence, the angle formed, by an arc of a circle equal in length to the radius of the circle, at its centre is one radian.
1 revolution = 360° = 2π radian
∴ 1 rad = \(\frac{180^{\circ}}{\pi}\) = 57.296°
As radian is the ratio of two lengths, it is dimensionless. It is only a number. For this reason the use of radian while measuring angles by any method is advantageous.
The angular displacement is a dimensionless physical quantity when measured in radian.
Polar Vector : Axial Vector
We know that linear displacement, velocity and acceleration are vector quantities.
Similarly, angular quantities like angular displacement, angular velocity and angular acceleration are also vectors. In order to express them completely, we need to mention their definite directions along with their magnitudes. By convention, the direc-tions of these vector quantities are taken along the axis of rota-tion.
Vectors like linear displacement, linear velocity, linear acceleration, linear momentum, force have real directions. These are known as polar vectors. The initial point of any polar vector is known as the pole of the vector.
On the other hand, the direction of the vectors associated with rotational motion (like angular displacement, angular velocity, angular acceleration, etc.) is imagined to be along a real axis, which is nothing but the axis of rotation. These vectors are called axial vectors.
Along the same circular path, the motion of a particle may be clockwise or anticlockwise. These two kinds of motion are opposite to each other. Hence, the two opposite directions of the axis of rotation are considered as the directions of the axial vectors in these two cases [Fig.]. The convention is: if the direction of rotation is along the direction of rotation of a right-handed screw then the direction of advancement of the screw-head indicates the direction of the axial vector.
For example, while opening the cap of a bottle placed on a table, if the rotation of the cap is anticlockwise (as seen from the above), then the advancement of the cap will be in the upward direction. This direction is chosen as the direction of the axial vectors like angular displacement. This is shown in Fig.(a). The opposite is shown in Fig.(b). The angular displacement Δθ along the circular path and the corresponding vector Δ\(\vec{\theta}\) along the axis of rotation are shown in the Fig.(c).
Relation between Linear Displacement and Angular Displacement
In Fig, let us assume that the length of arc AB is s and the radius of the circular path is r. Then, the angular displacement of the particle can be expressed as
i.e., s = rθ
or, distance travelled = radius × angular displacement
This distance s cannot be termed as the linear displacement of the particle, because it is a scalar quantity but displacement is a vector quantity. In circular motion, the direction of displacement changes continuously and hence, the magnitude of the distance travelled is not equal to the magnitude of displacement.
In the case of circular motion, the use of angular displacement is more advantageous than linear displacement for the following reasons:
i) Linear displacement changes direction continuously but the direction of angular displacement remains the same.
ii) Different particles of an extended body may travel different distances, but the angular displacement of every particle is the same. When an electric fan rotates, a particle at a greater distance from the centre of the fan covers more distance than a particle nearer to the centre, although both of them subtend equal angles at the centre of rotation, i.e., both have the same angular displacement.