A car moves uniformly along a horizontal sine curve y = a sin ( x / α ) , where a and α are certain constants. The coefficient of friction between the wheels and the road is equal to k . The maximum constant speed at which the car can travel without sliding is

Step-by-step solution

Step I : Determine the radius of curvature of given curve at point P (x, y). ∵ y = a sin x α ∴ d ⁡ y d ⁡ x = a α cos x α ∴ d ⁡ 2 y d ⁡ x 2 = - a α 2 sin x α ∴ the radius of curvature is R ⁡ = 1 + d ⁡ y d ⁡ x 2 3 / 2 d ⁡ 2 y d ⁡ x 2 R ⁡ = 1 + a 2 α 2 cos 2 x a 3 / 2 - a α 2 sin x α ...(i) Step II : Relate the friction and centripetal acceleration. In this case, centripetal acceleration is provided by force of friction. ∵ f ≤ f L or f ≤ k m g or mv 2 R ⁡ ≤ kmg or v ⁡ 2 R ⁡ ≤ kg ∴ v ⁡ max 2 R ⁡ min = k g ∴ v ⁡ max = k