A man is moving in a horizontal circular motion in the air. The man is tied by a rope that is angled at 60 to the vertical. If the man weighs 71 kg, find the tension of the rope in Newtons (N).

Question

BlackTom AI · Accepted Answer

1) Relevant knowledge points and concepts
- Resolve the rope tension T into vertical and horizontal components. When the rope makes an angle θ = 60° with the vertical, the vertical component is T cosθ and the horizontal component is T sinθ.
- The man has no vertical acceleration (he is in steady horizontal circular motion), so vertical forces must balance: T cosθ = mg, where m is mass and g is acceleration due to gravity (use g = 9.8 m/s^2).
- The horizontal component T sinθ provides the centripetal force, but its value is not needed to find T here.

2) Step-by-step reasoning
- Write the equilibrium condition in the vertical direction: T cos(60°) = m g.
- Solve this equation for T: T = m g / cos(60°).

3) Calculations with intermediate steps
- Compute the weight: m g = 71 kg × 9.8 m/s^2 = 695.8 N.
- Evaluate cos(60°) = 0.5.
- Divide: T = 695.8 N / 0.5 = 1391.6 N.

4) Why the correct answer is correct
- Because there is no vertical acceleration, the entire weight must be supported by the vertical component of the rope tension. With the rope at 60° to vertical, that component is T cos60; solving T cos60 = mg yields T = mg / 0.5, giving 1391.6 N. The horizontal component is determined by the circular motion but does not change the vertical balance that sets T.

5) Common incorrect approaches (why they are wrong)
- Using T sin60 = mg would be incorrect because sin60 is the component relative to the horizontal, not the vertical; that mistake gives a much larger T.
- Using g = 10 m/s^2 would yield 71×10/0.5 = 1420 N (approx), which is an approximation but not the value computed with g = 9.8.

6) Conclusion
- The tension in the rope is 1391.6 N, which matches the provided answer.

A man is moving in a horizontal circular motion in the air. The man is tied by a rope that is angled at 60 to the vertical. If the man weighs 71 kg, find the tension of the rope in Newtons (N).数值题

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