Motion of a car on banked road

The maximum permissible speed of a car on a banked rough road can be obtained as under.

Ncosθ = mg + fsinθ

Nsinθ + fcosθ = mv²/R 
          

If v = v_max then, f = f_max = μ_sN

On substituting then simplifying we get

v_max = {[(μ_s + tanθ)/(1 - μ_s tanθ)]Rg}¹^/²

Example 1:- A small block of mass m is moving over a smooth horizontal surface on a circular path at some moment. Length of string is L and velocity of block at some moment behind the tension in the string.

Solution: Here Tension provides, the required centripetal force

T = mv²/r = mvₒ²/L

Example 2:- Find the tangential acceleration of the bob of mass m at a moment when string makes angle θ with vertical as shown.

Solution: Let us make F.B.D. of bob, tension is radial, only force along tangent is

F_T = mgsinθ

a_T = F/m = gsinθ