Question

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. 

Answer 1

1. 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
   
   1. Weight mg acting vertically downwards
   2. Normal reaction N acting perpendicular to the road.
2. But in practice, vehicles never travel exactly with this speed.
3. 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.
4. 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
5. In this case, the direction of the force of static friction (f_s) between the road and the tires is directed along with the inclination of the road downwards.
6. The horizontal component (f_s 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)
7. The vertical component, N cosθ balances the component f_s sin θ and weight mg.
   ∴ N cos θ = f_s sinθ + mg
   ∴ mg = N cos θ −  f_s sin θ    ...(2) 
8. For maximum possible speed, f_s is maximum and equal to μ_sN. 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).
9. If µ_s = 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).
10. 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.