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5.2Torque and Rotational Dynamics

Connecting Linear and Rotational Motion — AP PHYSICS 1 Revision Notes & Topical Questions

5 min read 9 parts Lesson 9 of 18 1 quiz

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Lesson summary

5.2 Connecting Linear and Rotational Motion — AP PHYSICS 1 notes: Describe the linear motion of a point on a rotating rigid system from its rotational motion and vice versa.QuantityLinearAngularDisplacementsθVelocityvωAccelerationaαRigid body — every point has the same angular di…

  • Linear quantities
  • What the symbols mean
  • Centripetal acceleration
  • Not the same
  • Rolling without slipping
  • Belt or gear linkage without slipping

Section 1 of 9Linear and angular variables

QuantityLinearAngular
Displacementsθ
Velocityvω
Accelerationaα
Rigid body
  • Rigid body every point has the same angular displacement, velocity and acceleration.
Linear quantities
  • Linear quantities points farther from the axis have greater linear displacement, speed and tangential acceleration.

Section 2 of 9Arc length

s = rθ
  • s = rθ linear distance travelled by a point at distance r from a fixed axis, with θ in radians.
Radian
  • Radian the ratio of arc length to radius; dimensionless; 1 revolution = 2π rad = 360°.

Section 3 of 9Tangential speed

v = rω
  • v = rω from v = s/t = rΔθ/t.
Variation
  • Variation increases with distance from the axis, maximum at the rim, zero on the axis.
Direction
  • Direction tangent to the circle, so it is called tangential velocity.
Period
  • ω = 2π/T = 2πf.
    What the symbols mean: ω = angular frequency (rad s−1) · f = frequency (Hz) · T = period (s)

Section 4 of 9Tangential and centripetal acceleration

aT = rα
  • aT = rα from aT = Δv/Δt = rΔω/Δt, along the path.
Centripetal acceleration
  • Centripetal acceleration ac = v2/r = ω2r, toward the axis, present even at constant ω.
Total
  • Total a = √(at2 + ac2).
Not the same
  • Not the same angular acceleration and centripetal acceleration are different quantities.

Section 5 of 9Rolling and linked wheels

1Rolling without slipping
  • Rolling without slipping the contact point is instantaneously at rest, vcm = rω and acm = rα.
2Belt or gear linkage without slipping
  • Belt or gear linkage without slipping equal rim speeds, r1ω1 = r2ω2.

Section 6 of 9Converting between linear and angular quantities

1
Convert angles to radians and rev/min to rad/s (× 2π/60).
2
Identify the radius r of the point from the axis.
3
Use s = rθ, v = rω or aT = rα as required.
4
For rolling without slipping use the wheel radius with vcm = rω.
5
Check units: rad/s × m gives m/s.

Section 7 of 9More from the specification

1Rolling wheel dynamics on Earth
  • Rolling wheel dynamics on Earth rotational kinematics equations relate linear rim velocity to angular speed as cylinders roll down an Earth slope.
2Rotational kinematics
  • Equations linking Δθ, ω0, ω, α and t.
3Position
  • Location relative to a reference point.
4Vector
  • Quantity with magnitude and direction.

Section 8 of 9Key terms

TermMeaning
Arc lengths = rθ.
Tangential speedv = rω.
Tangential accelerationat = rα.
Centripetal accelerationac = v2/r = ω2r.
Rolling without slippingvcm = Rω.

Section 9 of 9Exam technique

1
Using degrees in s = rθ.
2
Giving different ω to points at different radii.
3
Forgetting centripetal acceleration at constant ω.
4
Confusing angular acceleration with centripetal acceleration.

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