Unit 3 · Topic 07

Circular-motion dynamics

Circular motion requires radial acceleration toward the centre. That acceleration does not create a new force: it must be produced by the radial component of the physical forces already present in the FBD.

Unit 3Circular dynamicsOpen navigation

Concept 01

Net radial force

Essential The minimum you should retain

For speed v on a circular path of radius R, radial acceleration has magnitude v²/R v^2/R and points toward the centre.

UnderstandInterpret and connect

The radial dynamical equation is ΣF_r = m a_r. If inward is chosen positive, it can be written ΣF_inward = m v²/R v^2/R .

DeepenFormulation and conditions

«Centripetal» describes the direction of acceleration or radial resultant; it is not an additional interaction. Tension, normal force, gravity, or friction may supply the radial force depending on the system.

ExploreConnections for further study

If speed changes, tangential acceleration may also exist. Newton's second law then separates naturally into radial and tangential directions.

Mathematical relation

Newton's second law in the radial direction

ΣFin=mv2R \sum F_{in}=m\frac{v^2}{R}
Represents

Newton's second law projected toward the centre of a circular path.

Physical interpretation

Centripetal describes the radial resultant, not a new interaction.

DeepenVariables, conditions, and checks

Variables

ΣF_in
inward radial resultant; usual unit: N
v
instantaneous speed; usual unit: m/s
R
local radius; usual unit: m

Conditions of application

  • The path is locally circular.
  • The radial convention is declared.

Dimensional check

kg·(m²/s²)/m=N.

Errors it helps prevent

  • Adding an extra centripetal force.
  • Treating radial direction as a fixed global axis.

Concept 02

Horizontal curve

Essential The minimum you should retain

On a flat unbanked curve, static friction between tyres and road may provide the horizontal force that keeps a vehicle on a circular path.

UnderstandInterpret and connect

For a horizontal road with no vertical acceleration, N = mg. The required friction is m v²/R v^2/R and must not exceed μ_sN if slipping is to be avoided.

DeepenFormulation and conditions

The condition m v²/R v^2/R ≤ μ_s mg gives v ≤ sqrt(μ_s g R) in this model. Mass cancels, so the ideal limit depends on μ_s, g, and R.

ExploreConnections for further study

This does not mean all real vehicles share the same limit. Tyres, aerodynamics, suspension, and load distribution can make the simple model inadequate.

Worked example

Maximum speed on an ideal flat curve

A flat curve has radius 50 m and static-friction coefficient 0.40.

Given
  • R = 50 m
  • μ_s = 0.40
  • g = 9.8 m/s²
Target

Find the maximum speed before sliding.

  1. Vertical balance

    N = mg.

  2. Radial direction

    Static friction supplies the required radial force m v²/R v^2/R .

  3. Threshold

    At the limit, μ_smg = m v²/R v^2/R .

  4. Calculation

    v_max = sqrt(μ_sgR) = sqrt(0.40·9.8·50) = 14.0 m/s.

Conclusion

The ideal maximum speed is about 14 m/s; mass cancels in this model.

Mathematical relation

Ideal speed limit on a flat curve

vmax=μsgR v_{max}=\sqrt{\mu_sgR}
Represents

The threshold speed before sliding in the simple flat-curve model.

Physical interpretation

Mass cancels in this ideal model.

DeepenVariables, conditions, and checks

Variables

v_max
ideal maximum speed; usual unit: m/s
μ_s
static coefficient; usual unit: dimensionless
R
radius; usual unit: m

Conditions of application

  • Flat unbanked curve.
  • Static friction supplies all horizontal radial force.
  • There is no vertical acceleration.
  • The Coulomb model is used.

Dimensional check

gR has dimension m²/s².

A car on a circular path has tangent velocity and inward radial friction in a separate FBD.carFBDtangent vinward f_s

Velocity is tangent and static friction points inward. No separate centripetal force is added.

Concept 03

Banked curve

Essential The minimum you should retain

A banked road tilts the normal force so that it has a horizontal component toward the centre of the curve.

UnderstandInterpret and connect

At one design speed, an ideal banked curve can use the horizontal component of N to supply the entire radial force without friction.

DeepenFormulation and conditions

With no friction and bank angle θ above horizontal: N cosθ = mg and N sinθ = m v²/R v^2/R . Dividing gives tanθ = v²/(Rg).

ExploreConnections for further study

If speed differs from the design value, friction may be needed and its direction depends on the tendency to slide up or down the bank. This is an extension, not a rule to memorize.

Mathematical relation

Ideal frictionless banked curve

tanθ=v2Rg \tan\theta=\frac{v^2}{Rg}
Represents

The relation between bank angle and design speed without friction.

Physical interpretation

The horizontal component of normal force supplies the radial resultant.

DeepenVariables, conditions, and checks

Variables

θ
bank angle; usual unit: rad or °
v
design speed; usual unit: m/s
R
radius; usual unit: m

Conditions of application

  • Friction is negligible.
  • The circular path is horizontal.
  • Vertical acceleration is zero.

Dimensional check

Both sides are dimensionless.

Errors it helps prevent

  • Assuming the angle depends on mass.
A banked road section shows angled normal force, vertical weight, and guides for normal-force components.vehicleNmg

Normal force is perpendicular to the road; its horizontal component points inward and its vertical component balances weight. The dashed guides separate those two components of the normal force.

Concept 04

Vertical circle

Essential The minimum you should retain

On a vertical circular path, the inward direction changes with position, so weight may help or oppose the required radial force.

UnderstandInterpret and connect

At the top, weight and an inward normal or tension can both contribute to m v²/R v^2/R . At the bottom, weight points opposite the inward radial direction.

DeepenFormulation and conditions

Write the local equation using a clear radial sign convention. At the bottom of a track with an upward normal force, for example, N - mg = m v²/R v^2/R .

ExploreConnections for further study

If a track can only push, N≥0 N\geq0 determines whether contact can be maintained. This block does not use energy to determine v; speed must be given or come from information within Unit 3 scope.

Top and bottom points show the local inward direction and normal and gravitational forces.inwardmg and contactinward; Nmgtopbottom

The inward direction changes: at the bottom it is upward and at the top downward. Radial equations are built locally without energy.

Concept review

Common errors

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

Adding a centripetal force in addition to tension, normal force, or friction.

«Centripetal» describes the required radial resultant, not an extra interaction.

Treating the radial direction as a globally fixed axis.

The inward direction changes with position along the path.