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NEET PHYSICSGRAVITATIONMedium

Question

A mass falls from a height hh and its time of fall tt is recorded in terms of time period TT of a simple pendulum. On the surface of the earth, it is found that t=2Tt=2T. The entire setup is taken on the surface of another planet whose mass is half of that of the Earth and whose radius is the same. The same experiment is repeated and corresponding times are noted as t′t' and T′T'. Then we can say:

A

t′=2T′t' = \sqrt{2}T'

B

t′>2T′t' > 2T'

C

t′<2T′t' < 2T'

D

t′=2T′t' = 2T'

Step-by-Step Solution

  1. Formulas:
  • Time of free fall from height hh: t=2hgt = \sqrt{\frac{2h}{g}} (derived from kinematic equation h=12gt2h = \frac{1}{2}gt^2). [Chapter 2, Example 2.4]
  • Time period of a simple pendulum: T=2πlgT = 2\pi\sqrt{\frac{l}{g}}. [Chapter 1, Example 1.5]
  • Acceleration due to gravity: g=GMR2g = \frac{GM}{R^2}. [Chapter 7, Eq. 7.9]

  1. Dependency on gg: Both tt and TT are inversely proportional to g\sqrt{g}. t∝1gandT∝1gt \propto \frac{1}{\sqrt{g}} \quad \text{and} \quad T \propto \frac{1}{\sqrt{g}}

  2. Ratio Analysis: The ratio of the time of fall to the time period is independent of gg: tT=2h/g2πl/g=2h2πl=constant\frac{t}{T} = \frac{\sqrt{2h/g}}{2\pi\sqrt{l/g}} = \frac{\sqrt{2h}}{2\pi\sqrt{l}} = \text{constant} Since hh and ll (length of pendulum) remain constant in the setup, this ratio remains constant regardless of the value of gg.

  3. Conclusion: On Earth, tT=2\frac{t}{T} = 2. On the other planet, despite the change in mass (M′=M/2M' = M/2) and resulting gravity (g′=g/2g' = g/2), the ratio must remain the same. t′T′=tT=2  ⟹  t′=2T′\frac{t'}{T'} = \frac{t}{T} = 2 \implies t' = 2T'

Exam Context & Concepts Covered

This question aligns with the NEET PHYSICS syllabus, specifically targeting concepts from GRAVITATION. 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.

PHYSICSGRAVITATIONheightrecordedperiodsimplependulum

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