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Circular Motion · JEE Reference

Banked Road & Friction

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3-D Banked Circular Road

⚠ Free Flight — Newton's 1st Law

x—

z—

dist from centre—

speed—

direction (θ from +x)—

vx—

vz—

No friction · No banking
Net horizontal force = 0
∴ constant velocity (N1L)

Trail thickness 1.0×

θ — Banking angle 20°

μ — Friction coeff 0.40

r — Radius (m) 50 m

m — Car mass (kg, doesn't affect skidding — see note) 1000 kg

v_min — friction acts UP slope

—

v_max — friction acts DOWN slope

—

Speed Explorer

v = m/s

v slider — m/s

Playback speed 1.0×

Enter a speed to see what happens

↙

v < v_min — Too slow

Gravity pulls car down the bank. Max static friction acting UP slope cannot prevent it. Car slides inward & down — settling at a smaller radius where v=v_min again, or off the inner kerb if the road isn't wide enough.

⚠

v = v_min — Limiting case

Car is just about to slide down. Friction is at its maximum: f = μN, acting up the slope.

✓

v_min ≤ v ≤ v_max — Safe zone

Static friction auto-adjusts to exactly what's needed. f < μN. No slipping — car stays on its circular path.

⚠

v = v_max — Limiting case

Car is just about to skid up the bank. Friction is at its maximum: f = μN, acting down the slope.

↗

v > v_max — Too fast

Required centripetal force exceeds what road can provide. Even max friction down slope is not enough. Car skids outward & up — settling at a wider radius where v=v_max again, or off the outer kerb if the road isn't wide enough.

—

Case 1 — High speed f down slope

Horizontal

\(N\sin\theta + f\cos\theta = \dfrac{mv^2}{r}\)

Vertical

\(N\cos\theta - f\sin\theta = mg\)

\(v_{\max}=\sqrt{\dfrac{rg(\sin\theta+\mu\cos\theta)}{\cos\theta-\mu\sin\theta}}\)

Case 2 — Low speed f up slope

Horizontal

\(N\sin\theta - f\cos\theta = \dfrac{mv^2}{r}\)

Vertical

\(N\cos\theta + f\sin\theta = mg\)

\(v_{\min}=\sqrt{\dfrac{rg(\sin\theta-\mu\cos\theta)}{\cos\theta+\mu\sin\theta}}\)

No friction (μ = 0)

Condition

\(\tan\theta=\dfrac{v^2}{rg}\)

Speed

\(v=\sqrt{rg\tan\theta}\)

Symbols

\(m\)	mass of car
\(v\)	speed
\(r\)	radius of curve
\(\theta\)	banking angle
\(\mu\)	static friction coeff.
\(N\)	normal reaction
\(f\)	friction (\(=\mu_s N\) at limit)

Note — mass cancels: \(N\propto m\) and \(f=\mu N\propto m\), so \(m\) divides out of every equation above. A loaded truck and an empty car skid at exactly the same v_min/v_max. The mass slider is kept for reference but doesn't change the physics here.

What happens past v_max/v_min: friction locks at \(\mu N\) and the car's actual radius is integrated from Newton's 2nd law (not animated by a fixed rule). It commonly settles at a new, slightly wider/narrower radius where v again equals the local v_max/v_min — exactly as a real skidding car finds a new line on a wide track. If that radius would be off the kerb, the car leaves the road and continues in a straight line (no more banking/friction acting on it).