AcademyApplying Force Models
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Dynamics of Circular Motion

Level 1 - Physics topic page in Applying Force Models.

Principle

Circular motion requires the net radial force to match the inward centripetal acceleration.

Notation

\(r\)
radius of curvature
\(\mathrm{m}\)
\(v\)
speed along the path
\(\mathrm{m\,s^{-1}}\)
\(\omega\)
angular velocity
\(\mathrm{rad\,s^{-1}}\)
\(a_c\)
centripetal acceleration magnitude
\(\mathrm{m\,s^{-2}}\)
\(\sum F_r\)
net force in the inward radial direction
\(\mathrm{N}\)
\(\theta\)
bank angle
rad or deg

Method

Derivation 1: Convert circular kinematics into a force equation

Earlier circular kinematics gave the inward acceleration magnitude. Newton's second law turns that acceleration requirement into a required inward resultant force.

Centripetal acceleration
\[a_c=\frac{v^2}{r}=r\omega^2\]
Radial second law
\[\sum F_r=ma_c\]
Take inward as the positive radial direction for this equation.
Radial force law
\[\sum F_r=m\frac{v^2}{r}=mr\omega^2\]

Derivation 2: Find the flat-curve speed limit

On a flat curve, the normal force and weight cancel vertically. Static friction is the only horizontal force available to point inward.

Friction supplies radial force
\[f_s=m\frac{v^2}{r}\]
Use the static limit
\[f_s\le\mu_sN=\mu_smg\]
Maximum speed
\[v_{\max}=\sqrt{\mu_sgr}\]

The free-body diagram below is a flat-curve model: weight and normal balance vertically, while static friction points inward and supplies the radial resultant.

carNWfs
Static friction is the inward resultant for a flat curve.

Derivation 3: Read a frictionless banked curve

For a frictionless bank, the normal force tilts. Its horizontal component supplies the radial force and its vertical component balances weight.

Vertical balance
\[N\cos\theta=mg\]
Radial balance
\[N\sin\theta=m\frac{v^2}{r}\]
Divide equations
\[\tan\theta=\frac{v^2}{rg}\]

Rules

These are the compact results from the method above.

Radial law
\[\sum F_r=m\frac{v^2}{r}\]
Angular form
\[\sum F_r=mr\omega^2\]
Flat curve limit
\[v_{\max}=\sqrt{\mu_sgr}\]
Banked curve
\[\tan\theta=\frac{v^2}{rg}\]

Examples

Question
A
\[900\,\mathrm{kg}\]
car takes a flat curve of radius
\[80\,\mathrm{m}\]
at
\[12\,\mathrm{m\,s^{-1}}\]
Find the required static friction.
Answer
Static friction supplies the radial force:
\[f_s=m\frac{v^2}{r}=900\frac{12^2}{80}=1620\,\mathrm{N}\]

Checks

  • The required radial resultant points inward.
  • A centripetal force is not a new force type; it is the inward resultant.
  • Constant speed still requires radial acceleration.
  • At the top of a hill or loop, inward is downward.
  • If the radial force is too small, the object cannot follow that circular path.

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Static and Kinetic Friction

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Fundamental Forces