A man is moving in a horizontal circular motion in the air. The man is tied by a rope that is angled at 300 to the vertical. The rope provides ..

Question

BlackTom AI · Accepted Answer

Tension in the rope has both a vertical component (which can balance weight) and a horizontal component (which provides the centripetal force).

1) Relevant concepts
- Tension is a single force along the rope; resolve it into perpendicular components.
- Vertical equilibrium: no vertical acceleration, so vertical component of tension equals weight (mg).
- Centripetal acceleration: horizontal component of tension supplies the required centripetal force m v^2 / r.

2) Step-by-step reasoning and equations
- Let T be the rope tension and the rope make angle 30° with the vertical.
- Vertical component: T_vertical = T cos 30°. Because the man is not accelerating vertically, T_vertical = mg.
  => T cos 30° = mg
  => T = mg / cos 30°.
- Horizontal component: T_horizontal = T sin 30°. Substitute T:
  => T_horizontal = (mg / cos 30°) * sin 30°.
  Using trig identity sin/cos = tan:
  => T_horizontal = mg * tan 30°.
- T_horizontal is the net horizontal force causing centripetal acceleration:
  => m a_c = m v^2 / r = T_horizontal = mg tan 30°.

3) Intermediate calculations shown
- cos 30° = √3/2, sin 30° = 1/2, tan 30° = 1/√3. So T = mg /(√3/2) = (2mg)/√3 and T_horizontal = mg*(1/√3).

4) Why the correct answer is correct
- The rope must both support the man against gravity (vertical component) and pull him inward to maintain horizontal circular motion (horizontal component). Hence it provides both centripetal force and cancels the weight.

5) Why other options are incorrect
- (a) Only cancel weight: ignores the needed horizontal centripetal force.
- (b) Only centripetal force: ignores vertical equilibrium so the man would fall.
- (d) No force: impossible, rope must exert tension to hold and guide the man.

Conclusion: The rope provides force for centripetal acceleration and to cancel out the weight of the man, so the correct choice is c.

A man is moving in a horizontal circular motion in the air. The man is tied by a rope that is angled at 300 to the vertical. The rope provides ..单项选择题

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