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NEET PHYSICSOscillationsMedium

Question

A particle is executing SHM along a straight line. Its velocities at distances x1x_1 and x2x_2 from the mean position are v1v_1 and v2v_2, respectively. Its time period is:

A

2πx12+x22v12+v222\pi \sqrt{\frac{x_1^2 + x_2^2}{v_1^2 + v_2^2}}

B

2πx22−x12v12−v222\pi \sqrt{\frac{x_2^2 - x_1^2}{v_1^2 - v_2^2}}

C

2πv12+v22x12+x222\pi \sqrt{\frac{v_1^2 + v_2^2}{x_1^2 + x_2^2}}

D

2πv12−v22x12−x222\pi \sqrt{\frac{v_1^2 - v_2^2}{x_1^2 - x_2^2}}

Step-by-Step Solution

  1. Velocity-Displacement Formula: The velocity vv of a particle performing Simple Harmonic Motion at a displacement xx from the mean position is given by: v=ωA2−x2v = \omega \sqrt{A^2 - x^2} where ω\omega is the angular frequency and AA is the amplitude.
  2. Formulate Equations: Squaring the relation gives v2=ω2(A2−x2)v^2 = \omega^2 (A^2 - x^2). Applying this to the two given cases: v12=ω2(A2−x12)— (i)v_1^2 = \omega^2 (A^2 - x_1^2) \quad \text{--- (i)} v22=ω2(A2−x22)— (ii)v_2^2 = \omega^2 (A^2 - x_2^2) \quad \text{--- (ii)}
  3. Eliminate Amplitude (AA): Subtract equation (ii) from equation (i) to eliminate AA: v12−v22=ω2(A2−x12)−ω2(A2−x22)v_1^2 - v_2^2 = \omega^2 (A^2 - x_1^2) - \omega^2 (A^2 - x_2^2) v12−v22=ω2(x22−x12)v_1^2 - v_2^2 = \omega^2 (x_2^2 - x_1^2)
  4. Solve for ω\omega: ω2=v12−v22x22−x12  ⟹  ω=v12−v22x22−x12\omega^2 = \frac{v_1^2 - v_2^2}{x_2^2 - x_1^2} \implies \omega = \sqrt{\frac{v_1^2 - v_2^2}{x_2^2 - x_1^2}}
  5. Calculate Time Period (TT): The time period is T=2πωT = \frac{2\pi}{\omega}. Substituting the expression for ω\omega: T=2πx22−x12v12−v22T = 2\pi \sqrt{\frac{x_2^2 - x_1^2}{v_1^2 - v_2^2}}

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.

PHYSICSOscillationsparticleexecutingstraightvelocitiesdistances

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