M018 · Where circular motion is possible

Banked Turn & Friction Envelope

Which speeds can this bank support without slipping? At one speed the inward component of the normal force supplies the turn without sideways friction. Slower motion needs uphill friction; faster motion needs downhill friction. Compare the required friction with its normal-force-dependent limit to find the allowed interval. Outside it, initial slip is inward at low speed and outward at high speed.

Tilt a circular road and drag the vehicle to change radius. Find the friction-free speed, feasible speed interval and initial slip direction.

Friction-free speed $12.7\,\mathrm{m\,s^{-1}}$

Circular-motion speed interval $\begin{gathered}v_{\min}=5.86\,\mathrm{m\,s^{-1}}\\v_{\max}=18.3\,\mathrm{m\,s^{-1}}\end{gathered}$

Normal / weight $1.1$

Friction reserve $0.35$

Model assumptions

This is an ideal point-mass assessment on a conical circular road, not road-design or driving advice. The radius is measured to the road centreline. The outer edge rises; positive friction points uphill and outward. The vehicle is enlarged twofold; forces act at its centreline reference point. Body height, wheel loads, torque balance, rollover, tyre dynamics and suspension dynamics are omitted. Normal and actual friction arrows add head to tail; the pale arrow is their sum, not a third contact force. All force arrows and the Coulomb cone share one linear adaptive scale. Static and initial sliding friction use the same ideal coefficient. Beyond the friction limit, the circular preview pauses and a red arrow shows initial cross-slope acceleration; no post-slip trajectory is integrated. Replay then inspects trial poses only. An infinite upper speed is a mathematical branch of this ideal law, not a real vehicle capability. Changing parameters preserves the current angle and restarts the trial without modeling the adjustment.

Start with the essentials

Focus question: Which speeds can this ideal road support without sideways slip?
One-sentence intuition: The allowed speed interval comes from the required friction staying within its normal-force-dependent limit. Slow and fast failures start slipping in opposite directions.

Core mathematical model

Define the contact directions

a_c=v^2/R,\quad N=m(g\cos\beta+a_c\sin\beta),\quad f_{\rm req}=m(g\sin\beta-a_c\cos\beta)

The outer edge is uphill. Positive friction points uphill/outward; the road normal points upward/inward. Forces refer to the centreline point mass.

The friction-free speed

v_0=\sqrt{Rg\tan\beta},\qquad |f_{\rm req}|\le\mu N

The contact cone has half-angle equal to the arctangent of the coefficient. Translated normal and friction arrows add head to tail with one common linear scale.

The lower and finite upper bounds

v_{\min}^2=Rg\max\!\left(0,\frac{\sin\beta-\mu\cos\beta}{\cos\beta+\mu\sin\beta}\right),\qquad v_{\max}^2=Rg\frac{\sin\beta+\mu\cos\beta}{\cos\beta-\mu\sin\beta}

The upper expression applies only when its denominator is positive. Otherwise the ideal Coulomb model has no finite upper bound; this is not a real vehicle speed claim.

Beyond the grip limit

f=\operatorname{clip}(f_{\rm req},-\mu N,\mu N),\qquad \ddot s=(f-f_{\rm req})/m

The cross-slope coordinate is positive uphill/outward. This is the initial acceleration at zero cross-slope velocity, using the same ideal coefficient for static and initial sliding friction. The preview pauses; no later slip trajectory is computed.

Common difficulties

Friction can point either way

Typical misconceptionFriction must always point inward on a bend.

Better mental modelAt low speed gravity tends to pull the mass downhill, so uphill friction is needed. At high speed the required friction points downhill/inward.

The contact sum is not an extra force

Typical misconceptionNormal, friction and their sum all act independently.

Better mental modelThe pale vector is just the sum of the two contact vectors. Add weight once to obtain the net force.

Run the experiment

01
Find the friction-free turn
Start with No sideways friction, then open Contact cross-section.
What to observe: The normal force alone supplies both the upward support and the inward acceleration.
02
Fail on each side
Compare Too slow with Too fast.
What to observe: The preview stops. The red arrow indicates opposite initial cross-slope accelerations.
03
Move through the envelope
Click the map or change angle and speed with the sliders.
What to observe: The marker, required-friction point and analytic speed interval update together. Mass changes force sizes but cancels from the ideal interval.