Unit 6 · Topic 02

Linear–angular relationships

Angular quantities are shared by the rigid body, while linear quantities depend on perpendicular distance from the axis.

Unit 6Linear–angular relationsOpen navigation

Concept 01

From angle to arc

Essential The minimum you should retain

For θ in radians, s=rθ s=r\theta and v=rω v=r\omega .

UnderstandInterpret and connect

At equal ω, the point farther from the axis travels a longer arc and moves faster.

DeepenFormulation and conditions

r is perpendicular distance to the axis, not any distance to the origin.

ExploreConnections for further study

Two points may share ω and have very different speeds.

Worked example

Two acceleration components at the rim

A point is 0.40 m from the axis with ω=5.0 rad/s and α=3.0 rad/s².

Given
  • r=0.40 m
  • ω=5.0 rad/s
  • α=3.0 rad/s²
Target

Find a_t, a_r, and |a|.

  1. Tangential

    a_t=rα a_t=r\alpha =1.20 m/s².

  2. Radial

    a_r=rω2 a_r=r\omega^2 =10.0 m/s².

  3. Geometry

    The components are perpendicular.

  4. Magnitude

    |a|=sqrt(1.20²+10.0²)=10.07 m/s².

Conclusion

a_t=1.20 m/s², a_r=10.0 m/s², and |a|≈10.07 m/s².

Mathematical relation

Arc–angle relations

s=rθ,v=rω s=r\theta,\quad v=r\omega
Represents

Arc and tangential speed of a point.

Physical interpretation

At larger radius, s and v are larger for the same rotation.

DeepenVariables, conditions, and checks

Variables

r
perpendicular distance to axis; usual unit: m
s
arc traveled; usual unit: m
v
tangential speed; usual unit: m/s

Conditions of application

  • θ is in radians.

Dimensional check

m and m/s.

Points at radii r and 2r compare tangential and radial vectors.v_Av_Ba_rOA (r)B (2r)

A and B share ω and α; because B is at twice the radius, its linear v, a_t, and a_r magnitudes are larger.

Concept 02

Changing speed

Essential The minimum you should retain

Tangential acceleration a_t=rα a_t=r\alpha changes velocity magnitude.

UnderstandInterpret and connect

It is tangent to the path and its sign follows the angular convention.

DeepenFormulation and conditions

For one α, it grows linearly with r.

ExploreConnections for further study

It may oppose v while the point slows.

Concept 03

Changing direction

Essential The minimum you should retain

Radial acceleration a_r=rω2 a_r=r\omega^2 =v²/r points toward the axis.

UnderstandInterpret and connect

It may be nonzero when ω is constant and α=0.

DeepenFormulation and conditions

Its quadratic dependence on ω makes it grow rapidly with rotation rate.

ExploreConnections for further study

It represents directional change, not inward speed.

Mathematical relation

Tangential and radial accelerations

at=rα,ar=rω2=v2r,a=at2+ar2 a_t=r\alpha,\quad a_r=r\omega^2=v^2/r,\quad a=\sqrt{a_t^2+a_r^2}
Represents

Linear acceleration components.

Physical interpretation

a_t changes speed and a_r changes direction.

DeepenVariables, conditions, and checks

Variables

a_t
tangential acceleration; usual unit: m/s²
a_r
radial acceleration; usual unit: m/s²

Conditions of application

  • Rotation about a fixed axis; r>0 for v²/r.

Dimensional check

m/s².

Velocity, tangential acceleration, and radial acceleration are shown at a rim point.va_ta_rrOP

v and a_t are tangent; a_r points toward the axis and changes the direction of v.

Concept 04

Perpendicular components

Essential The minimum you should retain

a_t and a_r are perpendicular in circular motion.

UnderstandInterpret and connect

Total magnitude is sqrt(a_t²+a_r²).

DeepenFormulation and conditions

They must be added vectorially, not as plain magnitudes.

ExploreConnections for further study

Both components vary with radius even when α and ω are shared.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Confusing a_t and a_r.

a_t changes speed; a_r changes direction.

Claiming constant ω means zero linear acceleration.

a_r may be nonzero.