Dynamics of Angular Motion

Dynamics of angular motion is the forces side of rotation: external torque changes angular momentum. For a rigid body rotating about a fixed axis with constant moment of inertia, the working equation is: ∑τₑₓₜ = Iα

1. Definitions (Must Know)

• Torque (moment of a force), τ (N m): turning effect of a force about a chosen point/axis.
  • Magnitude: τ = rF sin φ = Fd_⊥ where r is distance from axis to point of application, φ is the angle between vector r and vector F, and d_⊥ is the perpendicular moment arm.
• Net external torque, ∑τₑₓₜ: the signed sum of external torques about one chosen axis.
• Moment of inertia, I (kg m²): rotational “inertia” about an axis: I = ∑ mᵢrᵢ² for a set of discrete masses.
• Fixed-axis, constant-I form: ∑τₑₓₜ = Iα
• Symbols used in this lesson: τ torque (N m), F force (N), r radius (m), d_⊥ moment arm (m), φ angle (rad), I moment of inertia (kg m²), α angular acceleration (rad s⁻²).

2. Key Ideas (What Earns Marks)

• Always state the axis (or pivot) you’re taking moments about; torque depends on it.
• Use the perpendicular distance: τ = Fd_⊥ is often the safest form.
• I depends on both the mass distribution and the chosen axis.
• For a rigid body about a fixed axis: bigger I means smaller α for the same τₙₑₜ.

Torque formulas (choose the one that matches the diagram):

3. Detailed Explanations

A. Torque: what matters physically

For a given force magnitude F:

• torque increases if you apply it further from the axis (bigger r),
• torque increases if the force is more perpendicular to the radius (bigger sin φ),
• torque is zero if the line of action passes through the axis ( d_⊥ = 0).

B. Obtaining ∑τₑₓₜ = Iα

For a point mass m constrained at distance r from the axis, tangential acceleration is aₜ = rα. Its tangential resultant force is: Fₜ = maₜ = mrα The torque about the axis from this tangential force is: τ = rFₜ = r(mrα) = mr²α

For a rigid body, the general starting point is ∑ vector τₑₓₜ = d vector L/dt. When the body rotates about a fixed axis and I about that axis is constant, L = Iω, so:

This form is not valid unchanged when the rotation axis or moment of inertia varies; use the angular-momentum relation instead.

C. Choosing signs (1D rotation)

Pick a sign convention (e.g. anticlockwise is +). Then:

• torques that tend to rotate the body anticlockwise are positive,
• clockwise torques are negative, and you add them to get τₙₑₜ.

4. Common Mistakes

• Using rF when the force is not perpendicular (should be rF sin φ or Fd_⊥).
• Measuring r to the wrong point (use the distance from axis to where the force acts).
• Mixing torques about different axes in the same equation.
• Writing τ = Iω (incorrect; for a fixed axis and constant I, use ∑τₑₓₜ = Iα).

5. Exam Tips

• Draw the axis/pivot and mark d_⊥; it prevents many sign/geometry errors.
• In multi-force problems, compute each torque separately with its sign, then sum.
• Unit check:
  • I in kg m²,
  • α in rad s⁻²,
  • τ in N m.

6. Worked Examples

7. Mind Stretchers

8. Optional/Enrichment: Beyond Fixed-Axis Rotation

If the rotation axis changes direction (3D rigid body rotation) or the frame is non-inertial, you need more advanced tools (e.g. Euler’s equations / rotating-frame dynamics). This is beyond the fixed-axis H3 scope unless explicitly guided by the question.

Next step

Return to the Rotational Motion hub, or continue to the more general Torque and Angular Momentum relation.