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Questions
Obtain an expression for maximum safety speed with which a vehicle can be safely driven along a curved banked road.
Show that the angle of banking is independent of the mass of the vehicle.
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Solution
- The vertical section of a vehicle on a curved road (considering friction) of radius ‘r’ banked at an angle θ with the horizontal is shown in the figure.
If the vehicle is running exactly at the optimum speed, then the forces acting on the vehicle are
- Weight mg acting vertically downwards
- Normal reaction N acting perpendicular to the road.
- But in practice, vehicles never travel exactly with this speed.
- Hence, for speeds other than this, the component of the force of static friction between the road and the tires helps us, up to a certain limit.
- For maximum possible speed,
The component N sinθ is less than the centrifugal force `(mv^2)/r`.
∴ `(mv^2)/r` > N sinθ
Banked road: upper-speed limit - In this case, the direction of the force of static friction (fs) between the road and the tires is directed along with the inclination of the road downwards.
- The horizontal component (fs cos θ) is parallel to Nsinθ.
These two forces take care of the necessary centripetal force (or balance the centrifugal force).
∴ `(mv^2)/r = N sinθ + f_s cos θ` …(1) - The vertical component, N cosθ balances the component fs sin θ and weight mg.
∴ N cos θ = fs sinθ + mg
∴ mg = N cos θ − fs sin θ ...(2) - For maximum possible speed, fs is maximum and equal to μsN. From equations (1) and (2),
`v_max = sqrt(rg((tanθ + mu_s)/(1 - mu_stanθ)))` ...(3)
This is an expression for maximum safety speed with which a vehicle can be safely driven along a curved banked road (considering friction). - If µs = 0, then equation (3) becomes,
`v_max = sqrt(rg[(0 + tanθ)/(1 - 0tanθ)]`
∴ `v_max = sqrt(rg tanθ)` ...(4)
This is an expression of maximum safety speed with which a vehicle can be safely driven along a curved banked road (neglecting friction). - From equation (3) and equation (4), we can write,
`((tanθ + mu_s)/(1 - mu_s tanθ)) = V_max^2/(rg)` ...(5)
and `tan theta = V_max^2/(rg)`
∴ `theta = tan^-1(V_max^2/(rg))` ...(6)
From equation (5) and equation (6), angle of banking is independent of the mass of the vehicle.
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