A 20 kg girlslides down a frictionless slide at a playground. The surface of the slide is 8.0o to the horizontal. What is her acceleration? Give your answer to one decimal place.

Question

BlackTom AI · Accepted Answer

The component of gravity that causes acceleration down a frictionless incline is g·sinθ, so a = g sin(8.0°).

1) Relevant concepts:
- Gravity provides a weight force mg vertically downward.
- On an incline, resolve weight into components: perpendicular mg cosθ (balanced by normal force) and parallel mg sinθ (causes motion).
- Frictionless means no opposing frictional force. Newton’s second law along the incline: ΣF = ma applies with ΣF = mg sinθ.
- Mass cancels, so acceleration is independent of mass: a = g sinθ.

2) Step-by-step reasoning:
- Weight = mg (m = 20 kg, g ≈ 9.8 m/s^2).
- Parallel component down the slope = mg sinθ.
- Apply Newton’s second law along slope: mg sinθ = m a.
- Divide both sides by m to get a = g sinθ.

3) Calculations (all intermediate steps shown):
- θ = 8.0°. Compute sin(8.0°) ≈ 0.139173.
- Use g = 9.8 m/s^2 (standard approximation): a = 9.8 × 0.139173 ≈ 1.363 μm/s^2.
- Round to one decimal place: a ≈ 1.4 m/s^2.

(If using g = 9.81 m/s^2: a = 9.81 × 0.139173 ≈ 1.365 m/s^2 → 1.4 m/s^2 when rounded.)

4) Why this is correct:
- The only unbalanced force along the plane is mg sinθ, which sets the acceleration through a = (mg sinθ)/m = g sinθ. The small angle (8°) gives a small sine value, so acceleration is much less than g.

5) Why common incorrect alternatives are wrong:
- Using g cosθ (≈ 9.7 m/s^2) confuses the perpendicular component with the parallel component; cosθ is for normal force, not driving acceleration.
- Dividing by mass or inserting mass into the final formula (e.g., mg/ m incorrectly used) is redundant because mass cancels.
- Omitting the sine (using just g or θ in degrees directly) ignores the geometric projection of gravity along the slope.

6) Conclusion:
The correct acceleration is 1.4 m/s^2, which matches the provided answer.

A 20 kg girlslides down a frictionless slide at a playground. The surface of the slide is 8.0o to the horizontal. What is her acceleration? Give your answer to one decimal place.数值题

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