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NEET PHYSICSOscillationsMedium

Question

If T1,T2,T3,T4T_1, T_2, T_3, T_4 and T5T_5 represent the tension in the string of a simple pendulum when the bob is at the left extreme, right extreme, mean, any intermediate left and any intermediate right positions, respectively. Then, which of the following relations are correct?

(A) T1=T2T_1 = T_2 (B) T3>T2T_3 > T_2 (C) T4>T3T_4 > T_3 (D) T3=T4T_3 = T_4 (E) T5>T2T_5 > T_2

Choose the most appropriate answer from the options given below:

A

(A), (B) and (C) only

B

(B), (C) and (D) only

C

(A), (B) and (E) only

D

(C), (D) and (E) only

Step-by-Step Solution

  1. Tension Formula: The tension TT in the string of a simple pendulum of length LL and mass mm at an angular position θ\theta is given by T=mgcos⁡θ+mv2LT = mg\cos\theta + \frac{mv^2}{L}, where vv is the velocity of the bob.
  2. Analysis of Positions:
  • Mean Position (T3T_3): At the mean position (eta=0∘ eta = 0^{\circ}), velocity is maximum (vmaxv_{max}) and cos⁡0∘=1\cos 0^{\circ} = 1. Thus, tension is maximum: T3=Tmax=mg+mvmax2LT_3 = T_{max} = mg + \frac{mv_{max}^2}{L}.
  • Extreme Positions (T1,T2T_1, T_2): At the extreme positions (eta=etamax eta = eta_{max}), velocity is zero (v=0v=0). Thus, tension is minimum: T1=T2=mgcos⁡θmaxT_1 = T_2 = mg\cos\theta_{max}. Since the motion is symmetric, T1=T2T_1 = T_2. (Statement A is Correct).
  • Intermediate Positions (T4,T5T_4, T_5): At intermediate positions (0<eta<etamax0 < eta < eta_{max}), the velocity vv is non-zero (but less than vmaxv_{max}) and cos⁡θ<1\cos\theta < 1. The tension lies between the maximum and minimum values (Text<Tint<TmeanT_{ext} < T_{int} < T_{mean}).
  1. Evaluating Inequalities:
  • T3T_3 vs T2T_2: Since T3T_3 is maximum and T2T_2 is minimum, T3>T2T_3 > T_2. (Statement B is Correct).
  • T4T_4 vs T3T_3: T4T_4 is intermediate and T3T_3 is maximum. Thus, T4<T3T_4 < T_3. The statement T4>T3T_4 > T_3 is Incorrect (Statement C is False).
  • T5T_5 vs T2T_2: T5T_5 is intermediate tension and T2T_2 is minimum tension. Thus, T5>T2T_5 > T_2. (Statement E is Correct).
  1. Conclusion: The correct relations are (A), (B), and (E).

Exam Context & Concepts Covered

This question aligns with the NEET PHYSICS syllabus, specifically targeting concepts from Oscillations. Mastering this topic is crucial for scoring well in the upcoming medical entrance examinations. Solving conceptually related problems will help you understand the nuances of these concepts and improve your problem-solving speed.

PHYSICSOscillationsrepresenttensionstringsimplependulum

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