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(A) The kinetic energy $K$ of a particle in simple harmonic motion at a displacement $x$ is given by $K = \frac{1}{2} k(A^2 - x^2)$,where $A$ is the a

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$2\sqrt{6} \,cm$

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(A) The kinetic energy $K$ of a particle in simple harmonic motion at a displacement $x$ is given by $K = \frac{1}{2} k(A^2 - x^2)$,where $A$ is the amplitude.Maximum kinetic energy $K_{max} = \frac{1}{2} kA^2$.Given that at $x = 4 \,cm$,$K = \frac{1}{3} K_{max}$.Substituting the expressions: $\frac{1}{2} k(A^2 - 4^2) = \frac{1}{3} (\frac{1}{2} kA^2)$.Dividing both sides by $\frac{1}{2} k$: $A^2 - 16 = \frac{A^2}{3}$.Rearranging the terms: $A^2 - \frac{A^2}{3} = 16$.$\frac{2A^2}{3} = 16$.$A^2 = \frac{16 	imes 3}{2} = 24$.$A = \sqrt{24} = 2\sqrt{6} \,cm$.

For a particle executing simple harmonic motion,the kinetic energy of the particle at a distance of $4 \,cm$ from the mean position is $\frac{1}{3}$ of the maximum kinetic energy. The amplitude of the motion is

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