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Torque and Rotational Dynamics

5. Torque and Rotational Dynamics — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 8 parts Lesson 1 of 18 1 quiz

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Lesson summary

5. Torque and Rotational Dynamics — AP PHYSICS 1 notes: Describe rotation with angular kinematics, calculate torque and rotational inertia, and apply Newton's second law and static equilibrium to rotating systems.Rigid system — an object whose parts keep fixed distances from each…

  • Translation versus rotation
  • Angular variables
  • Sign convention
  • Average and instantaneous ω
  • ω = ω0 + αt
  • Δθ = ω0t + ½αt2

Section 1 of 8Rotational kinematics

Rotational kinematics
Rigid system
  • Rigid system an object whose parts keep fixed distances from each other.
Translation versus rotation
  • Translation versus rotation translation shifts the whole object;
    • rotation turns it about an axis, the line about which every point moves in a circle.
Angular variables
  • Δθ in radians , ω = Δθ/Δt and α = Δω/Δt.
    • (revolutions N = Δθ/2π)
Sign convention
  • Sign convention counterclockwise positive, clockwise negative; convert rpm and degrees to rad/s and radians.
Average and instantaneous ω
  • Average and instantaneous ω equal only when ω is constant.

Section 2 of 8Constant angular acceleration

1ω
ω0 + αt
  • velocity–time relation.
2Δθ
ω0t + ½αt2
  • displacement relation.
3ω2
ω02 + 2αΔθ
  • no time.
4Δθ
½(ω0 + ω)t
  • average velocity relation.
5Multi-stage motion
  • Multi-stage motion treat each stage separately, using final values as the next initial values.
6Benchmarks
  • Benchmarks ceiling fan about 5–10 rad/s, car wheel at highway speed about 80 rad/s, hard drive about 750 rad/s, dryer drum about 5 rad/s.

Section 3 of 8Linking rotation and linear motion

QuantityEquationDirection and effect
Arc lengths = rθRadians required
Tangential speedv = rωZero on the axis
Tangential accelerationat = rαAlong the motion; changes speed
Centripetal accelerationac = v2/r = ω2rToward the axis; changes direction
Total acceleration
  • √(at2 + ac2), as the components are perpendicular.
Constant ω
  • Constant ω a rotating point still has centripetal acceleration.
Frequency
  • Frequency if wheel A turns at twice the frequency of B, its ω and rim speed are twice as large.

Section 4 of 8Torque

1Torque
  • τ = rF sinθ = r⊥F = rF⊥, the ability of a force to change rotation.
2Lever arm
  • Lever arm r⊥ = r sinθ, the perpendicular distance from the axis to the line of action.
3Unit
  • N·m; dimensionally a joule but torque is not energy and is never written in joules.
4Force through the axis
  • Force through the axis produces zero torque.
5Sign
  • Sign counterclockwise positive, clockwise negative.
6Free-body diagrams of extended objects
  • Free-body diagrams of extended objects draw weight at the centre of mass, normal forces at contact points, tension at attachments, friction along surfaces.

Section 5 of 8Rotational inertia

Rotational inertia
I
Σmr2
  • resistance to angular acceleration, in kg·m2.
Dependence
  • Dependence on mass distribution and on the chosen axis.
    • (mass farther from the axis increases I)
Parallel-axis theorem
  • I = Icm + Md2, with d the distance between the parallel axes.
Systems
  • Systems add the rotational inertias of every part.

Section 6 of 8Newton's second law and equilibrium

Rotational second law
  • Στ = Iα; angular acceleration is proportional to net torque and inversely proportional to I.
Static equilibrium
  • ΣF = 0 and Στ = 0.
Balanced see-saw
  • Balanced see-saw m1r1 = m2r2.

Section 7 of 8Key terms

TermMeaning
Angular velocityRate of change of angular position, ω = Δθ/Δt.
TorqueRotational effect of a force, τ = rF sinθ.
Lever armPerpendicular distance from the axis to the line of action.
Rotational inertiaResistance to angular acceleration, I = Σmr2.
Parallel-axis theoremI = Icm + Md2.
Static equilibriumΣF = 0 and Στ = 0.

Section 8 of 8Exam technique

1
Using degrees or rpm in the rotational equations instead of radians and rad/s.
2
Using the object length for d in the parallel-axis theorem, or adding d instead of d2.
3
Applying the parallel-axis theorem when axes are not parallel or Icm is not about the centre of mass.
4
Writing torque units as joules.
5
Forgetting centripetal acceleration when α = 0.

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