Highway vehicle dynamics on a horizontal curve A vehicle of weight w (kgf equivalent) negotiates a horizontal circular curve of radius r. The centre of gravity is at height h above road level, and the distance between the wheel centres (track width) is 2a. Ignoring banking and tire slip, what is the maximum speed to avoid overturning about the outer wheels?

Introduction / Context:
In highway and vehicle dynamics, overturning on a level horizontal curve occurs when the lateral inertial effect shifts the resultant of weight and centrifugal force to pass through the outer wheel contact. This question asks for the limiting speed that prevents overturning, given geometry and center-of-gravity height.

Given Data / Assumptions:

• Vehicle weight = w (use m * g; it cancels in the ratio).
• Radius of curve = r.
• Height of centre of gravity above road = h.
• Track width = 2a (distance between wheel contact lines).
• Level road (no banking), no side slip; quasi-static analysis.

Concept / Approach:
At speed v, the lateral inertial force at the centre of gravity is m * v^2 / r, acting horizontally. Overturning about the outer wheels occurs when the overturning moment due to lateral force equals the restoring moment due to gravity. Use moment equilibrium about the outer wheel contact line to find the limiting velocity.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional check: g [m/s^2] * r [m] * a [m] / h [m] gives m^2/s^2 → square root is m/s (correct). Increasing h lowers v_max; increasing track half-width a or radius r raises v_max, which matches practical intuition.

Why Other Options Are Wrong:

• √(g r h / a): Inverts a and h; predicts higher v for higher CG—physically wrong.
• √(g a / (r h)) and √(g r / (a h)): Have incorrect dependence on r and a.
• g r a / h: Misses square root; wrong units.

Common Pitfalls:
Confusing overturning with skidding (the latter depends on friction). Also mixing up track half-width a with full track 2a. Always take moment about outer wheel contact for overturning on level curves.

Final Answer:
v_max = √(g r a / h)