A man is moving in a horizontal circular motion in the air. The man is tied by a rope that is angled at 730 to the vertical. If the man weighs 35 kg, find the tension of the rope in Newtons (N).

Question

BlackTom AI · Accepted Answer

Vertical equilibrium gives T cos 73° = mg, because the rope’s vertical component of tension must balance the man’s weight (there is no vertical acceleration).

1) Relevant concepts
- Weight W = mg (downward).
- Tension T in the rope has vertical component T cos θ (angle given to the vertical) and horizontal component T sin θ.
- Because the man moves in a horizontal circle, vertical forces balance: T cos θ = mg. The horizontal component provides centripetal force but is not needed to find T here.

2) Step-by-step reasoning and calculations
- Given: m = 35 kg, θ = 73° (to the vertical), use g = 9.8 m/s^2.
- Compute the weight: W = mg = 35 kg × 9.8 m/s^2 = 343 N.
- Set vertical equilibrium: T cos 73° = 343 N.
- Compute cos 73° ≈ 0.292372 (using a calculator).
- Solve for T: T = 343 N / cos 73° = 343 / 0.292372 ≈ 1173.3 N.

3) Intermediate steps shown
- W = 35 × 9.8 = 343 N
- cos 73° ≈ 0.292372
- T = 343 ÷ 0.292372 ≈ 1173.3 N

4) Why this is correct
Using the vertical component is correct because the rope is angled relative to the vertical; the vertical component T cos θ must equal the weight when there is no vertical acceleration. Using cos (not sin) is required because the angle is measured from the vertical.

5) Other possible mistakes (why they are wrong)
- Using T = mg (ignores rope angle) underestimates T.
- Using T sin 73° = mg (treats angle as from horizontal) is incorrect because the angle is given to the vertical.

Conclusion: Solving T cos 73° = 35·9.8 yields T ≈ 1173 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 730 to the vertical. If the man weighs 35 kg, find the tension of the rope in Newtons (N).数值题

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