A 965.8 kg car is driving in a roundabout on a frictionless banked track as shown in the diagram below.The angle of banking, [math: θ=14.8∘].What is the magnitude of the centripetal force acting on the car?*Hint: Break [math: FN] into components

Question

BlackTom AI · Accepted Answer

On a frictionless banked curve the horizontal component of the normal force provides the centripetal force. Using components: N sinθ = Fc and N cosθ = mg. Eliminating N gives Fc = mg tanθ.

1) Relevant concepts:
- Normal force components on a banked surface: vertical component balances weight, horizontal component gives centripetal force.
- Static vertical equilibrium: N cosθ = mg.
- Therefore Fc = N sinθ = (mg / cosθ) sinθ = mg tanθ.

2) Step-by-step reasoning and intermediate calculations:
- Mass m = 965.8 kg, take g = 9.81 m/s^2 so weight mg = 965.8 × 9.81 = 9474.498 N.
- Angle θ = 14.8°, compute tan(14.8°) ≈ 0.2642078.
- Multiply: Fc = mg tanθ = 9474.498 × 0.2642078.

3) Multiplication detail:
- 9474.498 × 0.2642078 ≈ 9474.498 × 0.2642078
- ≈ 2503.27 N (rounded to two decimal places).

4) Why this is correct:
- On a frictionless bank the only horizontal force available is the horizontal component of the normal force, so Fc must equal mg tanθ regardless of speed and radius for the geometry that produces no slipping. The algebra above directly follows from resolving N into components and eliminating N.

5) Why the other options are incorrect:
- b (9160.17 N) and c (12136.61 N) are much larger than mg tanθ and do not follow from the component relations; they do not satisfy vertical equilibrium N cosθ = mg while giving the stated horizontal component. 
- d (2420.22 N) is close but lower; it likely results from using a slightly different g or rounding in tanθ. The precise evaluation with g = 9.81 m/s^2 yields 2503.27 N.

Conclusion: The correct magnitude of the centripetal force is 2503.27 N, which matches option a.

A 965.8 kg car is driving in a roundabout on a frictionless banked track as shown in the diagram below.The angle of banking, [math: θ=14.8∘].What is the magnitude of the centripetal force acting on the car?*Hint: Break [math: FN] into componentsSingle choice

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