A car travels on a circular racetrack of radius 50 m, banked at angle θ. If the car travels at a speed 10 ms⁻¹, the wear and tear on its tyres is minimum. Taking g=10 ms⁻², the value of θ is:
- A.⁻¹((1)/(5))✓
- B.⁻¹((2)/(5))
- C.⁻¹((3)/(2))
- D.⁻¹(23)
Correct Answer
(A) ⁻¹((1)/(5))
Solution & Explanation
Tyre wear and tear comes from the friction (sideways) force between tyres and road. It is minimum when NO friction is needed, i.e. when the banking alone provides exactly the centripetal force. For a banked turn with no friction, the design condition is: tan θ = v²/(r g). Substitute v = 10 m s⁻¹, r = 50 m, g = 10 m s⁻²: tan θ = (10×10)/(50×10) = 100/500 = 1/5. Therefore θ = tan⁻¹(1/5), matching option A. Trap: at any other speed friction must act (inward at higher speed, outward at lower), producing wear — so the 'minimum wear' speed is precisely the no-friction design speed.
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