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(C) The total mechanical energy of the system is conserved because the surface is frictionless.At the maximum displacement (amplitude $A = 10 	ext{ c

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$67$

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(C) The total mechanical energy of the system is conserved because the surface is frictionless.At the maximum displacement (amplitude $A = 10 	ext{ cm} = 0.1 	ext{ m}$), the velocity is zero, so the total energy is purely potential energy: $E = \frac{1}{2} k A^2$.At any position $x$, the total energy is the sum of kinetic energy $(K)$ and potential energy $(U)$: $E = K + \frac{1}{2} k x^2$.Equating the two expressions for total energy: $\frac{1}{2} k A^2 = K + \frac{1}{2} k x^2$.Rearranging for $K$: $K = \frac{1}{2} k (A^2 - x^2)$.Given $K = 0.25 	ext{ J}$, $A = 0.1 	ext{ m}$, and $x = 0.05 	ext{ m}$:$0.25 = \frac{1}{2} k ((0.1)^2 - (0.05)^2)$.$0.5 = k (0.01 - 0.0025)$.$0.5 = k (0.0075)$.$k = \frac{0.5}{0.0075} = \frac{5000}{75} = \frac{200}{3} \approx 66.67 	ext{ Nm}^{-1}$.Rounding to the nearest integer, $k \approx 67 	ext{ Nm}^{-1}$.

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