The diagram below shows a 1.2 kg object being spun around in a circle from a 1.0 m long massless string. The angle [math: θ] is 37.9 degrees. The object is sped up such that the angle is 10 degrees larger. Determine the magnitude of the tension (N) in the string. Give your answer correct to 2 decimal places without units.

Question

BlackTom AI · Accepted Answer

1) Relevant knowledge points and concepts
For a conical pendulum (mass moving in uniform horizontal circle on a string making angle θ with the vertical) the tension T has two components: vertical T cosθ which balances the weight mg, and horizontal T sinθ which provides the centripetal force m v^2 / r. The vertical equilibrium relation T cosθ = mg is independent of the speed and is sufficient to find the magnitude of T. Use g = 9.81 m/s^2 (standard value).

2) Complete reasoning process step by step
The object is initially at θ = 37.9° and is sped up so the angle increases by 10°, giving the new angle θ' = 37.9° + 10° = 47.9°. Apply vertical equilibrium: T cosθ' = mg, so T = mg / cosθ'.

3) Calculations with intermediate steps
- New angle: θ' = 47.9°.
- Weight: mg = (1.2 kg)(9.81 m/s^2) = 11.772 N.
- Cosine: cos 47.9° ≈ 0.6704275 (calculator value).
- Tension: T = 11.772 / 0.6704275 = 17.556... N.
- Rounded to two decimal places: 17.56

4) Why the correct answer is correct
Because the vertical component of the tension must exactly balance the weight regardless of the horizontal circular motion, solving T cosθ = mg gives the exact tension for the new angle. No information about the speed is required once the angle and mass are known.

5) Common incorrect approaches and why they are wrong
- Using T sinθ = mg or T = mg tanθ is incorrect because the horizontal component (T sinθ) supplies the centripetal force, not the weight.
- Trying to compute T from centripetal expressions alone would require knowing the tangential speed v (or radius r and v), but T can be found directly from vertical balance.
- Using the original angle (37.9°) instead of the increased angle (47.9°) gives the wrong numeric tension.

6) Conclusion and confirmation
Following T = mg / cos(47.9°) with m = 1.2 kg and g = 9.81 m/s^2 yields T = 17.56, which matches the provided answer.

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