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.

Similar Questions

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).

Which of the following statements is False regarding the post-closing trial balance?

A closing entry may include a Blank ______.

Earnings per share is reported at the Blank ______ of the income statement.

Table 3 shows the short-run production costs for a car wash: [table] Costs | Cost per month ($) Land lease | 4,000 Workers’ wages | 2,600 Car wash machines | 3,000 Water | 1,500 Cleaning chemicals | 900 [/table] In the short run, calculate the car wash's fixed costs in Table 3.

During the closing process, the closing entry to reduce the sales revenue normal account balance to requires the revenue account be debited.

As of December 31, $2,500 of interest expense has accrued on a $50,000 note payable. The note payable and the accrued interest will become due and payable next year. How will the interest affect the adjustments at the end of the period?

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler