Unit 1 · Topic 06

Circular motion and relative velocity

Circular motion shows that speed and velocity are not equivalent. Relative velocity also requires precise identification of the frame from which an observation is made.

Unit 1Circular and relativeOpen navigation

Concept 01

Circular motion and radial acceleration

Essential The minimum you should retain

On a circle, instantaneous velocity is tangent to the trajectory. Radial or centripetal acceleration points toward the centre.

At constant speed, velocity is not constant because its direction changes continuously.

UnderstandInterpret and connect

Centripetal acceleration is not a new force. It names the radial component of acceleration; the responsible forces will be studied dynamically later.

The greater the speed or the smaller the radius, the greater the acceleration required to change direction.

DeepenFormulation and conditions

The radial acceleration vector can be written ar=−v2Rr^ a_r=-(v^2/R)\hat r . The negative sign indicates that it points opposite to the outward radial unit vector.

ExploreConnections for further study

On a curved path that is not circular, the normal direction still points toward the local centre of curvature. The radius of curvature may change from point to point.

Mathematical relation

Relation between linear speed and angular speed

v=ωR v=\omega R
Represents

Tangential speed on a circle of radius R.

Physical interpretation

For the same ω, a point farther from the axis covers more length per second.

DeepenVariables, conditions, and checks

Variables

v
tangential speed; usual unit: ms \mathrm{m/s}
ω
angular speed; usual unit: rad/s
R
trajectory radius; usual unit: m

Conditions of application

  • R is constant and ω is expressed in radians per unit time.

Dimensional check

(1/T)·L = LT L/T ; the radian is dimensionless.

Mathematical relation

Radial acceleration in circular motion

a→r=−v2Rr^,ac=v2R=ω2R \vec a_r=-\frac{v^2}{R}\hat r,\quad a_c=\frac{v^2}{R}=\omega^2R
Represents

The radial vector direction and centripetal magnitude.

Physical interpretation

The negative sign indicates a direction toward the centre relative to outward r^ \hat r .

DeepenVariables, conditions, and checks

Variables

aᵣ, ac a_{c}
radial acceleration and its magnitude; usual unit: ms2 \mathrm{m/s^2}
v
tangential speed; usual unit: ms \mathrm{m/s}
ω
angular speed; usual unit: rad/s
R
trajectory radius; usual unit: m

Conditions of application

  • The trajectory is circular with radius R at the instant considered.

Dimensional check

( LT L/T )²/L = LT2 L/T^2 .

Errors it helps prevent

  • Drawing acceleration tangent to the circle.
A particle is in the first quadrant of a circle. Its velocity is tangent and its acceleration points toward the centre.tangent varparticlecentre

Velocity is tangent to the circle. Radial acceleration is perpendicular to velocity and points toward the centre, even when speed is constant.

Concept 02

Circular motion with variable speed

Essential The minimum you should retain

If speed changes, acceleration has a tangential component in addition to the radial component.

The tangential component changes speed; the radial component changes direction.

UnderstandInterpret and connect

Total acceleration is the vector sum of two perpendicular components. Neither replaces the other.

DeepenFormulation and conditions

For a circle of fixed radius, at a_{t} = dvdt dv/dt and ar a_{r} = v2R v^2/R . Total magnitude is the square root of the sum of their squares.

ExploreConnections for further study

If speed and radius change simultaneously, both components may vary with time. Separating them reveals which part bends the path and which part changes the pace of motion.

Concept 03

Relative velocity and reference frames

Essential The minimum you should retain

vA/B v_{A/B} means the velocity of A measured from B. Subscript order is part of the definition.

The composition vA/C=vA/B+vB/C v_{A/C}=v_{A/B}+v_{B/C} connects three explicitly identified frames.

UnderstandInterpret and connect

Before adding velocities, draw or name which object observes which. Reversing the order reverses the sign: vB/A=−vA/B v_{B/A}=-v_{A/B} .

In this unit's classical approximation, time is common to the frames and velocities add vectorially.

DeepenFormulation and conditions

The composition relation is valid for classical frames. At speeds comparable with light, another transformation is required, beyond this unit's scope.

ExploreConnections for further study

Vector composition also describes boats in currents and aircraft in wind. In two dimensions, choosing one's own direction can compensate for a medium's velocity.

Mathematical relation

Classical composition of relative velocities

v→A/C=v→A/B+v→B/C \vec v_{A/C}=\vec v_{A/B}+\vec v_{B/C}
Represents

The velocity of A relative to C through an intermediate frame B.

Physical interpretation

The inner B subscripts form a chain and the result connects A to C.

DeepenVariables, conditions, and checks

Variables

vA/C v_{A/C}
velocity of A relative to C; usual unit: ms \mathrm{m/s}
vA/B v_{A/B}
velocity of A relative to B; usual unit: ms \mathrm{m/s}
vB/C v_{B/C}
velocity of B relative to C; usual unit: ms \mathrm{m/s}

Conditions of application

  • Galilean composition in classical frames; all vectors use compatible axes.

Dimensional check

Every term has dimension LT L/T .

Errors it helps prevent

  • Changing the order of the subscripts without reversing the vector.
Two relative velocities are placed head to tail and produce a resultant velocity between the first object and the final frame.vA/BvB/CvA/C

The head-to-tail construction represents vA/C=vA/B+vB/C v_{A/C}=v_{A/B}+v_{B/C} . The labels retain object and frame to avoid adding ambiguous quantities.

Concept review

Common errors

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

Confusing constant speed with constant velocity.

Speed can remain fixed while the direction of v changes continuously.

Calling centripetal force a new fundamental force.

Centripetal describes the radial direction of the required net force; the specific forces are studied in dynamics.

Drawing centripetal acceleration tangent to the path.

Velocity is tangent; radial acceleration points toward the centre.

Confusing the order of the subscripts.

Read vA/B v_{A/B} as the velocity of A observed from B. Reversing the order reverses the vector.

Adding velocities without identifying the frames.

Name the object and observer in each velocity and build a compatible chain of subscripts.