0.66 kg object attached to a 6.64 m string is moving in a circular motion on a vertical plane with 80 m/s uniform speed. What is the tension of the string at the highest point of the circle?

Question

BlackTom AI · Accepted Answer

At the top of the circle the centripetal acceleration is directed downward, so the sum of downward forces (tension T downward + weight mg downward) must provide the required centripetal force m v^2 / r. This gives the equation T + mg = m v^2 / r, so T = m v^2 / r − mg.

1) Relevant concepts:
- Centripetal force for uniform circular motion: F_c = m v^2 / r, directed toward the center.
- At the highest point in a vertical circle both gravity (mg) and the string tension act downward (toward the center).
- Use g = 9.8 m/s^2 unless otherwise specified.

2) Step-by-step reasoning and calculations:
- Given m = 0.66 kg, v = 80 m/s, r = 6.64 m.
- Compute v^2: v^2 = 80^2 = 6400 (m^2/s^2).
- Compute m v^2: m v^2 = 0.66 × 6400 = 4224 (kg·m^2/s^2).
- Compute centripetal requirement m v^2 / r: 4224 / 6.64 = 636.144578... N (this is the net downward force required).
- Compute weight mg: mg = 0.66 × 9.8 = 6.468 N (downward).
- Solve for tension: T = (m v^2 / r) − mg = 636.144578... − 6.468 = 629.676578... N.

3) Intermediate steps shown above; rounding to two decimal places gives T ≈ 629.68 N.

4) Why this is correct:
Because at the top the total downward force must equal m v^2 / r. Since both mg and T point downward there, T is the centripetal requirement minus the weight. The arithmetic confirms this directly.

5) Common incorrect approaches:
- Using T = m v^2 / r + mg (that would be correct at the bottom of the circle, not the top).
- Using g = 10 m/s^2 would give a slightly different numeric result.
- Forgetting to include mg or using wrong sign leads to incorrect tension.

Conclusion: The tension at the highest point is 629.68 N, which matches the provided answer.

0.66 kg object attached to a 6.64 m string is moving in a circular motion on a vertical plane with 80 m/s uniform speed. What is the tension of the string at the highest point of the circle?数值题

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