Angular Displacement

Angular displacement is an important concept in physics and mechanics that tells us how much an object has rotated around a fixed point or axis. Unlike linear displacement, which measures the straight-line distance between two positions, angular displacement focuses on the angle through which an object turns. It is usually measured in radians, degrees, or complete revolutions, making it essential for understanding rotational motion.Knowing about angular displacement helps us study the movement of objects like planets, wheels, gears, and other rotating systems. It also forms the foundation for other key concepts in physics, such as angular velocity, angular acceleration, and torque. Engineers, physicists, and students rely on this idea to analyze and solve real-world problems in areas like mechanics, robotics, and aerospace engineering.

1.0Definition of Angular Displacement

• The angle swept by the position vector of a moving particle during a given time interval is called its angular displacement.

• Angular displacement depends on the choice of origin but is independent of the reference line. As the particle moves along the circle, its angular position changes. If the particle rotates through an angle in time ( t ), then represents its angular displacement.

$$\begin{gathered} \text{Angular displacement}=\frac{\text{Arc}}{\text{Radius}}\Rightarrow \Delta \theta =\frac{s}{r} \\ \text{where s is arc length A to B.} \end{gathered}$$

2.0Units of Angular Displacement

• The SI unit of angular displacement is radian (rad).
• One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

In rotational motion, radians naturally relate linear distance traveled along a circular path to the radius: s = rθ

where s is arc length, r is radius, and θis angular displacement in radians.

• It is a dimensionless quantity

$(1,\text{radian}=\frac{360^{\circ }}{2\pi };\Rightarrow ;\pi ,\text{radian}=180^{\circ })$

3.0Sign Convention of Angular Displacement

Key Points:

• Angular displacement has magnitude and direction (vector quantity).
• Right-hand rule: Curl fingers in rotation direction; thumb shows positive axis direction.

4.0Key Points of Angular Displacement

• Clockwise angular displacement is taken as negative and anticlockwise displacement as positive.
• It is an axial vector and direction of angular displacement is perpendicular to the plane along the axis of rotation and is given by right hand thumb rule.
• Small angular displacement is a vector quantity,but large angular displacement is not a vector quantity, because it does not follow the commutative law of vector addition. ${\overrightarrow{d\theta }}_1+{\overrightarrow{d\theta }}_2={\overrightarrow{d\theta }}_2+{\overrightarrow{d\theta }}_1\text{ but }{\vec{\theta }}_1+{\vec{\theta }}_2\ne {\vec{\theta }}_2+{\vec{\theta }}_1$
• In one complete revolution angular displacement is 2 radian.
• If a body makes n revolutions, its angular displacement θ=2nπ radians  

Angular Displacement in Terms of Time

If the angular velocity of a rotating body is constant,the angular displacement in time t is,θ = ωt

For non-uniform rotation,

dθ = ω dt ⇒ θ = ∫ ω dt

Illustration-1.

A wheel rotates clockwise through 120° and then anticlockwise through  200° Find net angular displacement in radians using sign convention.
Solution:

$$\begin{gathered} \text{Sign Convention} \\ Anticlockwise\to PositiveClockwise\to Negative \\ {\theta }_1=-120^{\circ },{\theta }_2=+200^{\circ } \\ \text{Net angular displacement,}\theta ={\theta }_1+{\theta }_2=-120+200=+80^{\circ } \\ \text{In Radian} \\ \theta =80^{\circ }\times \frac{\pi }{180}=1.396\text{rad (anticlockwise)} \end{gathered}$$

Illustration-2.

Angular velocity of a rotating object is ω = 5 + 2 sin (2t) dt rad/s.Find angular displacement from t = 0 to t = π s
Solution:

$$\begin{gathered} \theta ={\int }_0^{\pi }(5+2\sin 2t)dt={\left[5t-\cos 2t\right]}_0^{\pi } \\ \theta =(5\pi -\cos 2\pi )-(0-\cos 0)=5\pi -1+1=5\pi \text{rad} \end{gathered}$$