A 905 kg car is driving in a roundabout on a frictionless banked track as shown in the diagram below.The angle of banking, [math: θ=15.4∘].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 centripetal force comes from the horizontal component of the normal force. Balance vertical forces to find FN, then take its horizontal component for Fc.

1) Relevant concepts
- Normal force FN acts perpendicular to the banked surface and can be resolved into vertical (FN cosθ) and horizontal (FN sinθ) components.
- Vertical equilibrium (no vertical acceleration): FN cosθ = mg.
- Horizontal component provides centripetal force: Fc = FN sinθ.
- Combining these gives Fc = (mg / cosθ) sinθ = mg tanθ.

2) Step-by-step reasoning
- Write vertical equilibrium: FN cosθ = mg → FN = mg / cosθ.
- Horizontal (centripetal) component: Fc = FN sinθ = (mg / cosθ) sinθ = mg tanθ.

3) Calculations (showing intermediate steps)
- m = 905 kg, g = 9.80 m/s^2, θ = 15.4°.
- Compute mg = 905 × 9.80 = 8869.0 N.
- Compute tan(15.4°). Using a calculator: tan(15.4°) ≈ 0.27578.
- Multiply: Fc = 8869.0 × 0.27578 ≈ 2445.42 N.

4) Why this is correct
The derivation uses force balance: because the surface is frictionless the only horizontal force available is FN sinθ. Using FN determined from vertical equilibrium yields Fc = mg tanθ. This directly produces the numeric result above.

5) Why the other options are incorrect
- Option d (≈2357.62 N) corresponds to mg·sinθ (the vertical component of weight projected onto the bank direction), not the horizontal centripetal component—incorrect.
- Option c (≈8559.29 N) is close to mg but not the horizontal centripetal component; it ignores the geometric tan factor.
- Option a (≈2823.96 N) does not follow from the correct mg·tanθ expression and is inconsistent with the resolved-force approach.

6) Conclusion
Using Fc = mg tanθ and plugging in the values gives Fc ≈ 2445.42 N, so the correct choice is (b) 2445.42 N, matching the provided answer.

A 905 kg car is driving in a roundabout on a frictionless banked track as shown in the diagram below.The angle of banking, [math: θ=15.4∘].What is the magnitude of the centripetal force acting on the car?*Hint: Break [math: FN] into components单项选择题

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