Angular Displacement

4.1.2 Angular Displacement

9. Definition of Angular Displacement

Angular displacement is the change in angular position of a rotating object.

Δθ = θ_f − θ_i

Where:

• θ_i is the initial angular position.
• θ_f is the final angular position.

10. Sign Convention

A common convention is:

• Anticlockwise rotation: positive
• Clockwise rotation: negative

Therefore:

Δθ > 0

for anticlockwise rotation, and:

Δθ < 0

for clockwise rotation.

The selected convention must remain consistent throughout the problem.

11. Angular Displacement as a Vector

Angular displacement has a direction along the axis of rotation.

Its direction is determined through the right-hand rule:

• Curl the fingers of the right hand in the direction of rotation.
• The extended thumb indicates the angular-displacement direction.

For rotation in a plane, this direction is perpendicular to the plane of rotation.

12. Radian

One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the circle’s radius.

If:

s = r

then:

θ = sr θ = 1 rad

13. Angular Unit Conversions

One complete revolution is:

1 rev = 360° = 2π rad

Therefore:

180° = π rad

Degree to radian conversion:

θ_rad = θ_degree × (π / 180)

Radian to degree conversion:

θ_degree = θ_rad × (180π)

Also:

1 rad ≈ 57.3°

14. Solved Example: Degree to Radian

Question: Convert 120° into radians.

Formula

θ_rad = θ_degree × (π / 180)

Solution

θ_rad = 120 × (π / 180) θ_rad = 2π / 3 θ_rad ≈ 2.09 rad

Final Answer

θ ≈ 2.09 rad

15. Arc Length

From:

θ = sr

Arc length is:

s = rθ

This equation must use θ in radians.

All particles in a rigid rotating body have the same angular displacement, but their arc lengths depend on their distances from the axis.

16. Diagram Placement

Rotational Kinematics

Keep its formulas available separately as searchable HTML:

s = rθ v = rω a_t = rα a_c = v²r = rω²