Oscillations

JEE Main level, single correct, medium

123 of 194

Question 123 of 194

A particle is making simple harmonic motion along the XX-axis. If at a distances x1{x}_{1} and x2{x}_{2} from the mean position the velocities of the particle are v1{v}_{1} and v2{v}_{2}, respectively. The time period of its oscillation is given as:

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Answer: (D) T=2πx22−x12v12−v22T=2\pi \sqrt{\dfrac{{x}_{2}^{2}-{x}_{1}^{2}}{{v}_{1}^{2}-{v}_{2}^{2}}}

v2=ω2(A2−x2){v}^{2}={\omega }^{2}\left({A}^{2}-{x}^{2}\right)
A2=x12+v12ω2=x22+v22ω2{A}^{2}={x}_{1}^{2}+\dfrac{{v}_{1}^{2}}{{\omega }^{2}}={x}_{2}^{2}+\dfrac{{v}_{2}^{2}}{{\omega }^{2}}
ω2=v22−v12x12−x22{\omega }^{2}=\dfrac{{v}_{2}^{2}-{v}_{1}^{2}}{{x}_{1}^{2}-{x}_{2}^{2}}
T=2πx12−x22v22−v12T=2\pi \sqrt{\dfrac{{x}_{1}^{2}-{x}_{2}^{2}}{{v}_{2}^{2}-{v}_{1}^{2}}}

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