Angular Displacement
WBBSE Class 11 Angular Displacement Notes
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 point A whose angular coordinate is θ1, and its final position is B whose angular coordinate is θ2. 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 displacement.
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, \(\angle A O B=\frac{\text { arc length } A B}{\text { radius } O A}\)
Actually, the relation, \(\theta=\frac{\text { arc }}{\text { radius }}\), provides the defination of an angle d. 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.
Read and Learn More: Class 11 Physics Notes
1 revolution =360° = 2π radian
∴ 1 rad = \(\frac{180^{\circ}}{\pi}=57.296^{\circ}\)
As radian is the ratio of two lengths, it is dimensionless. It is only a number. For this reason, the use of radians while measuring angles by any method is advantageous.
The angular displacement is a dimensionless physical quantity when measured in radians.
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Polar Vector Axial Vector: We know that linear displacement, velocity and acceleration are vector quantities.
Angular Displacement Formula and Derivation
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 directions of these vector quantities are taken along the axis of rotation.
- Vectors like linear displacement, linear velocity, linear acceleration, linear momentum, and 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.
- 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. The opposite is shown. The angular displacement Δθ along the circular path and the corresponding vector \(\Delta \vec{\theta}\) along the axis of rotation are shown.
Key Concepts of Angular Displacement
Relation Between Linear Displacement And Angular Displacement: 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
θ = \(\frac{\text { arc length } A B}{\text { radius } O A}=\frac{s}{r}\)
i.e., s = rθ
or, distance travelled = radius x 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.
Understanding Angular Displacement in Circular Motion
In the case of circular motion, the use of angular displacement is more advantageous than linear displacement for the following reasons:
- Linear displacement changes direction continuously but the direction of angular displacement remains the same.
- 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.