Rotational Kinematics

Rotational Kinematics

Textbook Subtopics

• 4.1.1 Angular Position
• 4.1.2 Angular Displacement
• 4.1.3 Angular Velocity
• 4.1.4 Angular Acceleration
• 4.1.5 Relationship Between Linear and Angular Kinematic Quantities

1. Learning Outcomes

After studying this topic, students should be able to:

• Define rotational motion and rotational kinematics.
• Explain the axis of rotation.
• Define angular position and angular displacement.
• Convert degrees and revolutions into radians.
• Calculate average and instantaneous angular velocity.
• Calculate angular acceleration.
• Apply the right-hand rule.
• Relate linear and angular quantities.
• Apply constant-angular-acceleration equations.
• Solve rotational-motion numerical problems.

2. Definition of Rotational Motion

Examples include:

• A rotating bicycle wheel
• A ceiling fan
• Earth rotating about its axis
• A spinning disc
• A rotating machine wheel

3. Rigid Body

A rigid body is an ideal object whose particles remain at fixed distances from one another during motion.

When a rigid body rotates:

• Every point moves in a circular path.
• The centres of these circles lie on the axis of rotation.
• All points sweep through the same angle in the same time.
• Points farther from the axis travel greater distances.

4. Axis of Rotation

The axis of rotation is the fixed line around which an object rotates.

It may pass through:

• A pivot
• A hinge
• A supporting shaft
• The centre of a wheel
• The object itself

Points lying directly on the axis have zero circular radius and do not move around the axis.

5. Rotational Kinematics

Definition

Rotational kinematics describes rotational motion without considering the forces and torques responsible for that motion.

It studies:

• Angular position
• Angular displacement
• Angular velocity
• Angular acceleration
• Time

6. Linear and Rotational Analogues

Linear and Rotational Analogues

36. Important Facts

• Radians must be used in s = rθ.
• One revolution equals 2π radians.
• Every point in a rigid body has the same angular velocity.
• Tangential speed increases with radius.
• Angular velocity points along the rotational axis.
• Tangential velocity acts along the tangent.
• Tangential acceleration changes speed.
• Radial acceleration changes velocity direction.
• A negative angular acceleration represents direction, not automatically slowing motion.

37. Key Logic

Rotational motion has the same mathematical pattern as translational motion:

s ↔ θ v ↔ ω a ↔ α

Linear and angular quantities are connected through radius:

s = rθ v = rω a_t = rα

38. Tips and Tricks

• Convert centimetres into metres before substitution.
• Convert rpm into rad s⁻¹.
• Use radians in all linear-angular relations.
• Choose an angular sign convention.
• Keep angular velocity and angular acceleration symbols separate.
• Use ω for angular velocity and α for angular acceleration.
• For rolling without slipping, distance travelled equals circumference multiplied by revolutions.
• Avoid rounding intermediate values.

39. Common Mistakes

• Using degrees directly in s = rθ.
• Confusing angular position with angular displacement.
• Writing angular velocity in m s⁻¹.
• Assuming different points have different angular velocities on a rigid body.
• Forgetting that tangential speed depends on radius.
• Directing angular velocity tangent to the circle.
• Mixing centripetal acceleration with tangential acceleration.
• Forgetting the negative sign during angular deceleration.
• Confusing rad s⁻¹ with rpm.

40. Remember Box

41. Exam Facts

• Angular position is measured relative to a reference line.
• Anticlockwise rotation is normally positive.
• Clockwise rotation is normally negative.
• Angular-quantity directions are determined by the right-hand rule.
• One revolution equals 2π rad.
• Rotational-motion equations require constant angular acceleration.
• v = rω gives tangential velocity.
• a_t = rα gives tangential acceleration.
• a_c = rω² gives radial acceleration.

42. Short-Question Preparation

Short-Question Preparation

43. Long-Question Preparation

Explain the relationship between linear and angular quantities.

Consider a point moving around a circle of radius r. It moves through arc length s. Its angular displacement is θ. When θ is measured in radians: s = rθ Differentiate this relation with respect to time. Since the radius is constant: ds/dt = r (dθ/dt) Linear velocity is v = ds/dt. Angular velocity is ω = dθ/dt. Therefore: v = rω Differentiate the velocity relation with respect to time: dv/dt = r (dω/dt) Tangential acceleration is at = dv/dt. Angular acceleration is α = dω/dt. Therefore: at = rα Radial acceleration acts toward the centre. It is given by: a_c = v²r = rω² Thus, radius connects linear and angular quantities.

44. MCQs

45. Sticky Notes

Sticky Notes

• 1 rev = 2π rad = 360°
• Angular displacement: Δθ = θ_f − θ_i
• Angular velocity: ω = Δθ / Δt
• Angular acceleration: α = Δω / Δt
• Arc length: s = rθ
• Tangential velocity: v = rω
• Tangential acceleration: a_t = rα
• Radial acceleration: a_c = rω²
• Use radians in linear-angular relations.

46. 60-Second Revision

θ = sr ω = dθ / dt α = dω / dt s = rθ v = rω at = rα ac = v²r = rω² For constant angular acceleration: ωf = ωi + αt Δθ = ωi t + ½ α t² ωf² = ω_i² + 2α Δθ

47. Topic Summary

Rotational kinematics describes angular motion without considering its causes. Angular position, displacement, velocity and acceleration correspond to linear position, displacement, velocity and acceleration. Radius connects both types through s = rθ, v = rω, and a_t = rα.